ENERGY STABLE HIGH-ORDER METHODS FOR SIMULATING UNSTEADY, VISCOUS, COMPRESSIBLE FLOWS ON UNSTRUCTURED GRIDS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY David M. Williams June 2013
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ENERGY STABLE HIGH-ORDER METHODS FOR SIMULATING UNSTEADY,
VISCOUS, COMPRESSIBLE FLOWS ON UNSTRUCTURED GRIDS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF
AERONAUTICS AND ASTRONAUTICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
David M. Williams
June 2013
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/tj082qz5745
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Antony Jameson, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Charbel Farhat
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Robert MacCormack
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
iii
Abstract
High-order methods have the potential to dramatically improve the accuracy and
efficiency of flow simulations in the field of computational fluid dynamics (CFD).
However, there remain questions regarding the stability and robustness of high-order
methods for practical problems on unstructured triangular and tetrahedral grids. In
this work, a new class of ‘energy stable’ high-order methods is identified. This class
of schemes (referred to as the ‘Energy Stable Flux Reconstruction’ class of schemes)
is proven to be stable for linear advection-diffusion problems, for all orders of ac-
curacy on unstructured triangular grids in 2D and unstructured tetrahedral grids
in 3D. Furthermore, this class of schemes is shown to be capable of recovering the
well-known collocation-based nodal discontinuous Galerkin scheme, along with new
schemes that possess explicit time-step limits which are (in some cases) more than 2x
larger than those of the discontinuous Galerkin scheme. In addition, the stability of
the Energy Stable Flux Reconstruction schemes is examined for nonlinear problems,
and it is shown that stability depends on the degree of nonlinearity in the flux and on
the placement of solution and flux points in each element. In particular, it is shown
that choosing the solution and flux point locations to coincide with the locations of
quadrature points promotes nonlinear stability by minimizing (or eliminating) non-
linear aliasing errors. A new class of symmetric quadrature points is identified on
triangles and tetrahedra for this purpose. Finally, the Energy Stable Flux Recon-
struction schemes and the new quadrature points are applied to several nonlinear
problems with the aim of assessing how well the schemes perform in practice.
iv
Acknowledgements
I would like to begin by thanking my adviser, Prof. Antony Jameson. Prof. Jameson’s
dedication to airplane design and the field of computational fluid dynamics has been
incredibly inspirational to me. I admire him a great deal for his passion and his
wealth of expertise. During my time with him, he has been very generous in sharing
his wisdom and advice, both professional and personal in nature. It has been an
honor to work alongside him for the last 4 years.
I would like to thank Prof. Peter Vincent and research scientist Patrice Castonguay
for supervising my research and serving as mentors. Peter has been very influential
in encouraging my research interests in high-order methods. During his time with the
lab, he did a great job of leading by example, working very hard to publish results and
to raise grant money for the lab. His efforts are greatly appreciated, and will not soon
be forgotten. Patrice has also significantly aided my research. He was very patient
and persistent in solving the many challenges, both analytical and computational in
nature, that frequently arose during the course of my research. I admire him for his
composure and the quick-thinking that he displayed throughout our time together.
I would like to thank Manuel Lopez who served as a co-teaching assistant and research
collaborator. His hard work in organizing lab events, preparing course work, and
performing research has been invaluable. I am incredibly grateful for his assistance.
He has been like a brother to me, and I will miss the good times that we had while
hanging out both inside and outside of lab.
I would like to thank research collaborators Prof. Guido Lodato and research scientist
Lee Shunn. It has been a pleasure working alongside them, and I greatly appreciate
their contributions, and their patience with my many questions.
I would like to thank Professors Juan Alonso, Sanjiva Lele, Robert MacCormack,
Charbel Farhat, Michael Saunders, Parviz Moin, Philipp Birken, and Brian Cantwell
for their contributions to my education and my research at Stanford. Each of them
has had an important influence on my understanding of numerical methods and com-
pressible fluid mechanics.
I would like to thank the National Science Foundation and the Stanford Graduate
v
vi
Fellowship (SGF) for their financial support. In particular, I would like to thank the
Morgridge family, who directly funded my Ph. D. research through the SGF program.
I am forever grateful for their generosity.
I would like to thank Andre Chan for proof-reading large parts of my thesis. His
diligent efforts in this regard have been extremely valuable.
I would like to thank my ‘first generation’ of lab mates: Edmond Chiu, Matt Culbreth,
Lala Yi, Yves Allaneau, Kui Ou, and Aniket Aranake. Thank you for welcoming me
to the lab, and providing me with your assistance with research and homework. I will
always remember the fun that we had in classes, and at seminars and conferences.
I would like to thank my ‘second generation’ of lab mates: David Manosalvas, Ab-
hishek Sheshadri, Kartikey Asthana, Jerry Watkins, Joshua Romero, Cyrus Liu,
Mehul Oswal, and Alex Fickes. Their enthusiasm for computational fluid dynam-
ics has been inspirational to me. I greatly enjoyed our times doing research together,
and our frequent visits to Panda Express, Tree House, Coho, and the Axe and Palm.
Finally, I would like to thank my faith, friends, and family. Without each of them,
none of this would have been possible. In particular, I would like to thank my Lord and
Savior Jesus Christ. I would like to thank my friends Adam Nicholl, Colin Miranda,
Grady Chang, Timothy Williams, Ethan Kung, Cory Combs, Matt Tang, Eric Chu,
It is generally believed that higher fidelity numerical methods are needed for solving
many of the unsteady, compressible, viscous flow problems that arise in industrial
settings. Currently, such problems are frequently solved using 2nd-order finite vol-
ume (FV) methods [1, 2]. These methods are well-suited for simulating a wide range
of compressible, viscous, steady flow problems, including (most notably) high-speed
flows around aircraft flying at cruise conditions [3, 4, 5]. These methods are par-
ticularly robust for steady problems because they produce appreciable amounts of
numerical dissipation that tend to dampen spurious oscillations, thereby encouraging
the flow to converge towards a steady state. Furthermore, the implementation of these
methods is straightforward on unstructured meshes of triangular and tetrahedral el-
ements, making them well-suited for simulating flows around complex geometries for
which it may not be possible to create structured cartesian meshes. However, despite
these advantages, 2nd-order methods encounter significant challenges when they are
applied to unsteady flows, including flows around aircraft that are ascending, de-
scending, or experiencing gust, buffet, or flutter phenomena [6]. Figure (1.1) shows
examples of the unsteady flow phenomena that arise during the ascent and descent
of civilian aircraft. In this setting, the numerical dissipation produced by 2nd-order
methods tends to obscure important time-dependent flow features, including vortices
and low-amplitude acoustic waves [9, 10]. The current inability, in practice, to reliably
predict these unsteady flow phenomena contributes to the creation of unnecessarily
conservative aircraft designs, that, in turn, drive up the cost of flight. As a result,
there has been significant interest in developing high-order methods that are suitable
for unstructured meshes (cf. the recent review article by Vincent and Jameson [11]
and the recently assembled collection of articles edited by Wang [12]). For many of
the unsteady flow problems that arise in practice, these high-order methods have the
potential to produce less dissipation and to obtain more accurate results at lower
computational cost [13].
The potential advantages of using high-order methods in place of 2nd-order methods
can be illustrated by performing numerical experiments on the classic ‘vortex propa-
gation problem’ originally formulated by Hu et al. [14] and Gassner et al. [15]. This
CHAPTER 1. INTRODUCTION 3
Figure 1.1: Examples of unsteady flow phenomena that arise during the ascent and descentof commercial jets. The top and bottom photographs were taken by Daniel Umana [7] andVik Sridharan [8], respectively. The photographs are under copyright by their respectivephotographers, and have been reprinted here with permission.
problem involves the propagation of an inviscid, isentropic vortex in a quiescent fluid.
In this scenario, the vortex propagates indefinitely, and the exact solution can be
straightforwardly computed from the initial conditions. In 3D, the exact solution of
this problem takes the following form
ρ = ρ0
(1− γ − 1
2Λ2
) 1
γ−1
, (1.1)
ρu = ρ (u0 + rxc0 Λ) , (1.2)
ρv = ρ (v0 + ryc0 Λ) , (1.3)
ρw = ρ (w0 + rzc0 Λ) , (1.4)
E =p0
γ − 1
(1− γ − 1
2Λ2
) γγ−1
+ρ
2
(u2 + v2 + w2
), (1.5)
CHAPTER 1. INTRODUCTION 4
where
c0 =
√γp0ρ0
, (1.6)
Λ = Λmax exp
1−
(|r|r0
)2
2
, (1.7)
r = r× (x− x0 − u0 t) , (1.8)
and where ρ is the density, γ is the ratio of specific heats, u = (u, v, w) is the
velocity vector, E is the total energy, p is the pressure, c is the speed of sound, Λ
characterizes the strength of the vortex, r0 is the radius of the vortex, r = (rx, ry, rz)
and r = (rx, ry, rz) are the orthogonal orientation vectors for the vortex, x = (x, y, z)
is the position vector for the vortex, and t is time. Here, it should be noted that all
quantities subscripted by 0 denote values at initial time t0.
In order to obtain a numerical solution to the problem defined by equations (1.1) –
(1.8), it was necessary to approximately solve the Euler equations. The Euler equa-
tions are the partial differential equations (PDEs) that govern the behavior of inviscid,
compressible fluid flows. It should be noted that these equations can be obtained by
setting the viscosity µ equal to zero in the Navier-Stokes equations (cf. Chapter 8).
In a series of numerical experiments performed by the author, these equations (the
Euler equations) were approximately solved on a cuboid domainΩ = [−5, 5]×[−5, 5]×[−5, 5] using a 2nd-order FV method and 3rd and 4th-order discontinuous Galerkin
(DG) methods. In general, the truncation error for the methods was order p+1, where
p denoted the order of the polynomial basis used to represent the solution in each
element. For each method, the vortex propagation problem with initial conditions
and x0 = (0, 0, 0) was solved on a series of tetrahedral grids formed by splitting
cartesian grids withN = N3 hexahedral elements into grids withN = 6N3 tetrahedral
elements. The explicit 5-stage, 4th-order Runge-Kutta scheme of Carpenter and
Kennedy [16] was used to advance the solution in time (starting from t0 = 0), and
the time-step was chosen small enough to ensure that temporal errors were negligible
relative to spatial errors. The spatial errors were computed at t = 2.5 seconds using
a broken L2 norm of the difference between the exact and approximate solutions.
CHAPTER 1. INTRODUCTION 5
In order to minimize contributions to the error from the boundary conditions, exact
boundary conditions were imposed and the error was evaluated inside a box centered
around the vortex (with a domain defined by −1.25 ≤ x ≤ 1.25, −1.25 + t ≤ y ≤1.25 + t, −5 ≤ z ≤ 5). Table (1.1) shows the spatial error (in the energy E) on the
grids with N = 8, 16, 32, and 64. Results are tabulated for the 2nd-order FV scheme
with p = 1 and the DG methods with p = 2 and 3. Based on the tabulated data, it
is clearly evident that (as expected) the high-order methods are more accurate than
the 2nd-order FV method for a given mesh resolution N .
Table 1.2: Normalized execution times of the 2nd-order FV scheme (with p = 1) and thehigh-order DG schemes (with p = 2 and p = 3) for the vortex propagation problem.
From these results, it is evident that the high-order methods are capable of producing
the same level of accuracy as the 2nd-order FV method while requiring less compu-
tation time. In particular, although the case with N = 8 and p = 3 and the case
with N = 32 and p = 1 produce roughly the same amount of error, results from the
case with N = 8 and p = 3 can be obtained approximately 19 times faster. This is
CHAPTER 1. INTRODUCTION 6
expected, as according to theory, the high-order methods (with p = 2 and p = 3)
produce significantly less dissipation than the 2nd-order method (with p = 1), and
thus accurately preserve the vortex structure for a longer period of time.
Based on these numerical experiments, it is clear that high-order methods have the
potential to outperform 2nd-order methods on unsteady flow problems. However, the
potential of high-order methods has yet to be fully realized in practice. For many
of the complex, nonlinear flow problems that arise in industrial settings, high-order
methods appear to be less robust and more difficult to implement than their 2nd-
order counterparts. Therefore, despite their promise, they have yet to be adopted by
the majority of fluid dynamicists.
1.2 Efforts to Improve High-Order Methods
There have been efforts to improve the flexibility and ease of implementation of high-
order methods, and in particular the well-known DG methods. These endeavors have
brought about a rise in popularity of high-order methods that omit the explicit use
of quadratures, allowing them to serve as simpler alternatives to the classical DG
methods that use a ‘method of weighted residuals’ formulation that requires complex
quadrature procedures (as described in [17, 18, 19]). In particular, various unstruc-
tured high-order schemes based directly on differential forms of the governing system
have emerged, such as ‘nodal’ DG methods [20, 13], Spectral Difference (SD) meth-
ods [21, 22] and, more generally, Flux Reconstruction (FR) methods [23]. For conve-
nience, such schemes will henceforth be referred to as ‘FR type’ schemes. FR type
schemes do not require integration to be performed, and hence their implementations
can (explicitly at least) omit quadratures. As such, efficient implementation of FR
type schemes is straightforward relative to their more traditional counterparts. Con-
sequently, it is envisaged that such FR type methods will become popular amongst a
wide community of fluid dynamicists within both academia and industry.
In what follows, a review of FR type methods is provided in order to establish the con-
text and motivation for the current work. First proposed by Huynh in 2007, FR is an
approach which generates new high-order schemes and recovers well-known schemes,
including a variety of collocation-based nodal DG and SD schemes [23]. The FR ap-
proach should not be confused with the classical concept of flux reconstruction that
CHAPTER 1. INTRODUCTION 7
appears in the context of traditional 2nd-order FV schemes (cf. [24, 25, 26, 27, 28]),
its closely related adaptations (cf. [29, 30, 31]), or the more recently developed flux
reconstruction procedures for formulating a posteriori error estimations for DG meth-
ods (cf. [32, 33, 34]). In the FR approach [23], the flux is subject to a reconstruction
procedure involving ‘correction functions,’ which are required to be polynomials of
one degree higher than the solution, as well as satisfying symmetry and boundary
constraints. The FR approach was originally formulated for advection problems in
1D and was extended via tensor products to quadrilateral elements in 2D [23]. In
2009, Huynh formulated an extension of the FR approach to diffusion problems [35]
in which both the solution and the flux are subject to reconstruction procedures. In
both [23] and [35], Fourier analysis was employed in an effort to evaluate the stabil-
ity of various FR schemes for linear advection and diffusion problems. Thereafter,
Huynh developed an extension of the FR approach to advection-diffusion problems
on triangles [36]. This approach makes use of scalar-valued, 2D correction functions,
which are required to satisfy symmetry conditions similar to those imposed on the 1D
correction functions. To the author’s knowledge, Fourier analysis of these schemes
has yet to be performed.
In 2009, Gao and Wang identified a closely related class of schemes for advection-
diffusion problems, referred to as Lifting Collocation Penalty (LCP) schemes [37, 38].
These schemes make use of ‘weighting functions’ which are different from correction
functions in that they are piecewise continuous polynomials that are of one degree
lower than the correction functions. The LCP approach involves multiplying the
governing equations by the weighting functions and integrating the result in order
to obtain ‘corrections’. These corrections are similar in form to those that arise
in Huynh’s FR approach [23, 35], and under certain circumstances in 1D, the two
approaches can be shown to be identical [39]. As a result, Wang, Gao, Haga, and
Yu have referred to the class of all FR and LCP schemes as ‘Correction Procedure
via Reconstruction’ (CPR) schemes [40, 39], although it is not yet clear whether all
FR schemes are in fact LCP schemes or vice versa. Wang, Gao, and Haga have
provided evidence for the stability of the schemes, successfully applying the FR and
LCP schemes to nonlinear advection-diffusion problems on grids of quadrilaterals and
triangles in 2D [37] and tetrahedra and prisms in 3D [41, 40].
In addition to being related to the LCP schemes, the FR schemes are also related
CHAPTER 1. INTRODUCTION 8
to the Summation By Parts Simultaneous Approximation Term (SBP-SAT) finite
difference schemes. The SBP-SAT and FR formulations are similar in that the SBP-
SAT formulation can recover the 2nd-order Galerkin Finite Element scheme on tri-
angles [42] which (for a linear solution basis) is akin to the collocation-based nodal
DG approach that the FR formulation recovers. However, SBP-SAT schemes of the
form given by [42] cannot recover Galerkin Finite Element methods for non-simplex
elements [43], and (thus far) have not been extended to yield compact high-order dis-
cretizations on triangles. For this reason, the SBP-SAT schemes will not be discussed
further, although the interested reader should consult [44, 45, 46, 47] for details per-
taining to these schemes. The remainder of this discussion will instead focus on FR
type schemes which are more closely related to the schemes of Huynh [23, 35, 36],
and Gao and Wang [37, 38].
Recently, efforts have focused on rigorously proving the stability of FR type schemes
using ‘energy methods’. Such methods offer advantages over von Neumann techniques
(i.e. Fourier analysis) since they automatically extend to all orders of accuracy, and
are valid for unstructured grids. In 2010, Jameson [48] used an energy method to prove
stability of a particular SD method for linear advection problems. Subsequently, in
2011, Vincent, Castonguay, and Jameson [49] used a similar approach to derive an
entire class of stable FR schemes for linear advection problems in 1D. These stable
schemes, referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes, are
parameterized by a single scalar. The scalar influences the analytical form of the
correction functions, and variations of the scalar lead to recovery of various known
numerical methods, including a collocation-based nodal DG method, the SD method
that Jameson [48] proved to be stable for linear advection, and Huynh’s so-called g2
method [23]. Recently, Castonguay and Williams et al. [50, 51] extended the VCJH
schemes, and proved their stability for advection-diffusion problems in 1D [51].
In 2D, Castonguay, Vincent, and Jameson used an energy method to identify a class
of VCJH schemes which they proved to be stable for linear advection problems on
triangles [52]. These schemes make use of vector-valued, 2D correction functions,
which are required to satisfy symmetry and orthogonality conditions. The VCJH
schemes on triangles are parameterized by a single scalar (as in the 1D case) which,
if set to the correct value, allows for recovery of the collocation-based nodal DG
scheme [52].
CHAPTER 1. INTRODUCTION 9
Note that the symmetry conditions used to define the VCJH correction functions due
to Castoguay et al. [52] are different than those used to define the correction functions
due to Huynh [36], and thus it does not appear that these two classes of schemes are
equivalent. In addition, because VCJH schemes due to Castonguay et al. [52] utilize
correction functions and LCP schemes due to Gao and Wang [38] utilize weighting
functions, the equivalence of the VCJH and LCP classes of schemes has yet to be
demonstrated.
In summary, a number of authors have proposed FR type approaches for the treatment
of advection and advection-diffusion problems in 1D and in higher dimensions on
triangles and tetrahedra. However, for linear advection problems, only the VCJH
approaches of [49] in 1D and [52] on triangles have been proven stable for all orders
of accuracy. Furthermore, for linear advection-diffusion problems on triangles, the
approach of [52] has not yet been proven stable. This thesis will extend the approach
of [52], to provide for the first time on triangles and tetrahedra, a provably stable
family of FR schemes for linear advection-diffusion problems.
The format of this thesis is as follows. Part 1, Chapters 2 – 5, describe the procedure
and linear stability theory associated with the energy stable FR schemes for advection-
diffusion problems. In particular, Chapter 2 presents an overview of the energy stable
FR schemes in 1D, Chapter 3 presents an overview of the energy stable FR schemes
on triangles and proves the stability of the schemes for linear advection-diffusion
problems in 2D, Chapter 4 presents an overview of the energy stable FR schemes
on tetrahedra and proves the stability of the schemes for linear advection-diffusion
problems in 3D, and finally, Chapter 5 presents a reformulation of the energy stable
FR schemes on triangles from Chapter 3, utilizing the theoretical advancements from
Chapter 4.
Part 2, Chapters 6 and 7, describe the nonlinear stability theory associated with
the energy stable FR schemes for advection-diffusion problems. In particular, Chap-
ter 6 discusses how to minimize aliasing errors that arise when the energy stable
FR schemes are applied to nonlinear advection-diffusion problems in 3D, and Chap-
ter 7 presents a new class of solution point locations that minimize aliasing errors,
where the solution point locations are set to coincide with quadrature point locations
associated with a new class of quadrature rules.
CHAPTER 1. INTRODUCTION 10
Finally, Part 3, Chapters 8 and 9, describe the results of numerical experiments that
involve applying the energy stable FR schemes to nonlinear problems. In particular,
Chapter 8 describes the nonlinear governing equations for fluid flow (the Navier-Stokes
equations), and Chapter 9 presents the results of experiments that involve applying
the schemes from Part 1 and the solution point locations from Part 2 to a range of
fluid dynamics problems.
Part I
Stability Theory for Linear
Advection-Diffusion Problems
11
Chapter 2
Energy Stable Flux Reconstruction
for Advection-Diffusion Problems in
1D
12
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 13
2.1 Preamble
In order to provide a suitable context for the energy stable FR schemes (VCJH
schemes) on triangles in 2D and tetrahedra in 3D, this section presents a review
of the energy stable FR schemes in 1D.
2.2 Preliminaries
The important aspects of the energy stable FR schemes in 1D can be conveniently il-
lustrated via application of the schemes to the advection-diffusion equation. Consider
the 1D advection-diffusion equation which takes the following form
∂u
∂t+
∂
∂x
(f(u,∂u
∂x
))= 0, (2.1)
where u denotes the scalar solution, f denotes the scalar flux, and (as discussed
previously) x and t denote the spatial and temporal coordinates, respectively.
In following the standard approach of Cockburn and Shu [53], one may introduce an
auxiliary variable q into equation (2.1) as follows
∂u
∂t+
∂
∂x(f (u, q)) = 0, (2.2)
q − ∂u
∂x= 0. (2.3)
Through this operation, the 2nd-order PDE (equation (2.1)) is reformulated as a
system of 1st-order PDEs (equations (2.2) and (2.3)).
Solutions to equations (2.2) and (2.3) are sought in the 1D domain Ω with boundary
Γ. In following the conventional Finite Element approach, Ω can be divided into N
non-overlapping elements such that
Ω =
N⋃
k=1
Ωk, (2.4)
Ωi ∩Ωj = ∅ ∀i 6= j. (2.5)
Approximate forms of equations (2.2) and (2.3) can be constructed on the kth element
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 14
(Ωk). In particular, the solution u in equation (2.2) can be approximated by a function
uDk = uDk (x, t) which is required to be a polynomial of degree p within the kth element
and to assume the value of zero outside of the kth element. The polynomial portion
of uDk can be constructed as follows
uDk =
Np∑
i=1
(uDk)iℓi (x) , (2.6)
where(uDk)i
is the value of the approximate solution at solution point (‘nodal point’)
i, and ℓi (x) is the Lagrange polynomial of degree p which takes on the value of 1 at
solution point i and the value of 0 at all other solution points. Figure (2.1) shows an
example of the Np = p + 1 = 3 solution points that reside in the element Ωk for the
case of p = 2.
xk xk+1
Ωk
Figure 2.1: An example of the Np = 3 solution points (denoted by circles) that reside withinthe domain of the kth element Ωk for the case of p = 2.
In equation (2.6), the superscript D on uDk indicates that the approximate solution
may be discontinuous in the following sense: the sum of uDk and uDk+1, defined on
neighboring elements Ωk and Ωk+1, is not required to reside in C0 (Ωk ∪Ωk+1), i.e.
the solution is not required to be continuous at the interfaces between elements.
In a similar fashion, the flux in equation (2.2) can be approximated by a function
fDk = fD
k (x, t) which is required to be a polynomial of degree p within the kth element
and to assume the value of zero outside of the kth element. Evidently, the polynomial
portion of fDk can be constructed as follows
fDk =
Np∑
i=1
(fDk
)iℓi (x) , (2.7)
where(fDk
)iis the value of the approximate flux at solution point i. As before, the
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 15
superscript D on fDk indicates that the approximate flux may be discontinuous at the
interfaces between elements.
An approximate form of equation (2.3) can be obtained in the same way. In particular,
the solution u can be approximated (as before) by uDk , and the auxiliary variable q
can be approximated by a function qDk = qDk (x, t) which has the same properties (with
regard to degree and continuity) as uDk and fDk . As a result, the polynomial portion
of qDk can be defined as follows
qDk =
Np∑
i=1
(qDk)iℓi (x) , (2.8)
where(qDk)i
is the value of the approximate auxiliary variable at solution point i.
On rewriting equations (2.2) and (2.3) in terms of uDk , fDk , and qDk (defined in equa-
tions (2.6), (2.7), and (2.8)), one obtains the following
∂uDk∂t
+∂fD
k
∂x= 0, (2.9)
qDk −∂uDk∂x
= 0. (2.10)
It turns out that, in their current form, equations (2.9) and (2.10) do not comprise
a valid numerical scheme. For example, in equation (2.9), the solution in the kth
element (uDk ) depends entirely on quantities in the kth element, and there is no ex-
change of information with neighboring elements. In order to enable communication
between elements (and to allow for conservation of u), it is necessary to introduce
a continuous flux fk into equation (2.9) and to introduce a continuous solution uk
into equation (2.10). In particular, one may define fk as a degree p + 1 polynomial,
where the sum of fk and fk+1 is required to be in C0 (Ωk ∪Ωk+1), and one may define
uk as a degree p + 1 polynomial, where the sum of uk and uk+1 is required to be in
C0 (Ωk ∪Ωk+1). Upon replacing fDk with fk in equation (2.9) and replacing uDk with
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 16
uk in equation (2.10), one obtains
∂uDk∂t
+∂fk∂x
= 0, (2.11)
qDk −∂uk∂x
= 0. (2.12)
Equations (2.11) and (2.12) do not comprise a complete numerical scheme (as the
quantities fk and uk have yet to be precisely defined), however, they do comprise a
valid scheme in the sense that they allow for the exchange of information between
neighboring elements.
In what follows, the FR procedure for solving equations (2.11) and (2.12) will be
described. However, in order to enable a convenient description of this procedure,
it is first necessary to transform equations (2.11) and (2.12) from the domain of the
physical element Ωk = x | xk ≤ x ≤ xk+1 to the domain of the reference element
ΩS = x | − 1 ≤ x ≤ 1. Towards this end, one may define the following mapping
between the coordinates in Ωk and ΩS
x = Θk (x) =
(1− x2
)xk +
(1 + x
2
)xk+1. (2.13)
This mapping has a Jacobian denoted by Jk that is defined as follows
Jk =dΘk (x)
dx=
1
2(xk+1 − xk) . (2.14)
The mapping and its Jacobian can be utilized to transform the quantities uDk , fk,
qDk , and uk from physical space into the equivalent quantities uD, f , qD, and u in
reference space, as follows
uD = Jk uDk (Θk (x) , t) , f = fk (Θk (x) , t) , (2.15)
qD = J2k q
Dk (Θk (x) , t) , u = Jk uk (Θk (x) , t) . (2.16)
Upon substituting uDk , fk, qDk , and uk (as defined in equations (2.15) and (2.16)) into
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 17
equations (2.11) and (2.12), one obtains
∂uD
∂t+∂f
∂x= 0, (2.17)
qD − ∂u
∂x= 0. (2.18)
The energy stable FR procedure for solving equations (2.17) and (2.18) is described
in the next section.
2.3 FR Procedure
The energy stable FR procedure for solving advection-diffusion problems consists of
eight stages. However, it should be noted that the procedure can be described in
seven stages if one elects to combine the 3rd stage and the very brief 4th stage,
(cf. Chapter 3).
The first stage involves computing the discontinuous solution in the reference space,
which as mentioned previously, is denoted by uD. In order to compute this quantity,
one may multiply both sides of equation (2.6) by the Jacobian Jk, and substitute
equations (2.13) and (2.15) into the result in order to obtain the following
uD =
Np∑
i=1
(uD)iℓi (Θk (x)) =
Np∑
i=1
(uD)iℓi (x) , (2.19)
where(uD)i
= Jk(uDk)i
is the value of the discontinuous solution in the reference
space at solution point i, and ℓi (x) is the Lagrange polynomial in the reference space
that assumes the value of 1 at solution point i and the value of 0 at all other solution
points. Figure (2.2) shows an example of the approximate solution uD for the case of
p = 2.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 18
ΩS
uD
Figure 2.2: An example of the approximate solution uD defined on the reference element ΩS
for the case of p = 2.
The second stage involves computing common values of the discontinuous solution at
the flux points located on the edges of the reference element. In 1D, the reference
element has two flux points, where one flux point is located on the left edge of the
element (at x = −1) and the other flux point is located on the right edge of the
element (at x = 1), as shown in Figure (2.3).
ΩS
−1 1
Figure 2.3: The flux points (denoted by squares) on the edges of the reference element ΩS .
The discontinuous solution in the reference space (equation (2.19)) can be evaluated
at the flux points on the left and right edges in order to obtain uDL = uD (−1) and
uDR = uD (1). Figure (2.4) shows an example of the approximate solution values uDLand uDR for the case of p = 2.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 19
ΩS
uD
L
uD
R
Figure 2.4: An example of the solution values uDL and uDR defined on the left and right edgesof the reference element ΩS for the case of p = 2.
Next, uDL and uDR which reside in the reference space can be transformed into the
quantities uDL− and uDR− which reside in the physical space (utilizing equation (2.15)),
where the minus sign subscripts signify that these quantities belong to the physical
domain of the kth element. A similar procedure can be employed to compute uDL+
and uDR+, where the plus sign subscripts signify that these quantities belong to the
physical domains of the nearest neighbors of the kth element. Figure (2.5) shows an
example of the approximate solution values uDL−, uDR−, u
DL+, and u
DR+ for the case of
p = 2.
ΩkΩk−1 Ωk+1
uD
L−
uD
L+
uD
R−
uD
R+
Figure 2.5: An example of the solution values uDL−, uDR−, u
DL+, and u
DR+ defined on the left
and right edges of Ωk for the case of p = 2.
Thereafter, a common value of the solution at the flux point on the left edge (denoted
by u⋆L) can be computed from uDL− and uDL+, and a common value of the solution at
the flux point on the right edge (denoted by u⋆R) can be computed from uDR− and uDR+.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 20
Figure (2.6) shows an example of the common solution values u⋆L and u⋆R for the case
of p = 2.
ΩkΩk−1 Ωk+1
u⋆
L
u⋆
R
Figure 2.6: An example of the solution values u⋆L and u⋆R defined on the left and right edgesof Ωk for the case of p = 2.
Finally, in preparation for the next stage, the common solution values u⋆L and u⋆R in
the physical space can be transformed to the reference space, yielding u⋆L and u⋆R,
in accordance with equation (2.15). Figure (2.7) shows an example of the solution
values u⋆L and u⋆R for the case of p = 2.
ΩS
u⋆
R
u⋆
L
Figure 2.7: An example of the solution values u⋆L and u⋆R defined on the left and right edgesof the reference element ΩS for the case of p = 2.
The third stage involves forming the continuous solution in the reference space (u)
by adding a ‘solution correction’ (uC) to the discontinuous solution in the reference
space (uD). In order to ensure that u is continuous, the sum of uC and uD is required
to equal the common solution values u⋆L and u⋆R at the flux points on the left and
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 21
right edges of the reference element, respectively. This requirement can be satisfied
by defining uC as follows
uC =(u⋆L − uDL
)gL (x) +
(u⋆R − uDR
)gR (x) , (2.20)
where gL (x) and gR (x) are ‘correction functions’ that are required to satisfy the
following point-wise constraints
gL (−1) = 1, gL (1) = 0, (2.21)
gR (−1) = 0, gR (1) = 1, (2.22)
and the following symmetry constraints
gL (x) = gR (−x) , gL (−x) = gR (x) . (2.23)
Figure (2.8) shows an example of uC for the case of p = 2.
ΩS
uC
Figure 2.8: An example of uC defined on the reference element ΩS for the case of p = 2.
Before proceeding further, it is important to discuss in more detail the correction
functions that are used to construct uC. There are an infinite number of such correc-
tion functions (which satisfy equations (2.21) – (2.23)), however, it is well-known that
not all of them yield stable FR schemes. This has driven efforts to identify a class of
‘energy-stable’ correction functions that are guaranteed to preserve the stability of the
FR schemes. The ‘VCJH correction functions’ comprise one such class of correction
functions. The VCJH correction functions are exceptional, in that they are required
to satisfy equations (A.1) – (A.4) in Appendix A, in addition to equations (2.21) –
(2.23). In order to satisfy these equations, the correction functions gL (x) and gR (x)
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 22
are required to be defined by equations (A.5) and (A.6). The VCJH definitions of
gL (x) and gR (x) are parameterized by a scalar constant κ, where it is important to
note that an infinite family of correction functions can be obtained by varying the
value of κ. Figures (2.9)–(2.11) show examples of the correction functions gL (x) and
gR (x) for different values of κ.
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(a) gL, κ = κdg
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(b) gR, κ = κdg
Figure 2.9: Examples of the correction functions gL and gR for κ = κdg, for the case ofp = 2, where κdg is defined in Appendix A.
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(a) gL, κ = κsd
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(b) gR, κ = κsd
Figure 2.10: Examples of the correction functions gL and gR for κ = κsd, for the case ofp = 2, where κsd is defined in Appendix A.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 23
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(a) gL, κ = κ+
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
(b) gR, κ = κ+
Figure 2.11: Examples of the correction functions gL and gR for κ = κ+, for the case ofp = 2, where κ+ is defined in Appendix A.
Choosing gL (x) and gR (x) to be VCJH correction functions is beneficial because
(under a few minor assumptions) this ensures that the resulting FR schemes are
stable for linear advection-diffusion problems in 1D (as shown in [51]).
Now, having defined uC and the associated correction functions, one may construct
the continuous solution in the reference space (u) by summing uD and uC as follows
u = uD + uC = uD +(u⋆L − uDL
)gL (x) +
(u⋆R − uDR
)gR (x) . (2.24)
Figure (2.12) shows examples of u and uk for the case of p = 2, where uk =
J−1k u
(Θk (x)
−1) in accordance with equation (2.16).
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 24
ΩS
u
(a) Example of u
ΩkΩk−1 Ωk+1
uk
uk+1
uk−1
(b) Example of uk
Figure 2.12: Examples of u and uk defined on the reference element ΩS and the physicalelement Ωk (respectively) for the case of p = 2.
The fourth stage involves computing the discontinuous auxiliary variable in the refer-
ence space (denoted by qD). Towards this end, one may substitute u (equation (2.24))
into equation (2.18) and rearrange the result in order to obtain
qD =∂uD
∂x+(u⋆L − uDL
) dgLdx
+(u⋆R − uDR
) dgRdx
=
Np∑
i=1
(uD)i dℓi
dx+(u⋆L − uDL
) dgLdx
+(u⋆R − uDR
) dgRdx
. (2.25)
Figure (2.13) shows an example of qD for the case of p = 2.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 25
ΩS
qD
Figure 2.13: An example of qD defined on the reference element ΩS for the case of p = 2.
The fifth stage involves computing the discontinuous flux in the reference space (de-
noted by fD). Towards this end, one may compute(fDk
)i, the value of the flux in the
physical space at each solution point i, using the values of(uDk)i
and(qDk)i
which can
be obtained by evaluating equations (2.19) and (2.25) at each solution point i, and
transforming the result to the physical space via equations (2.15) and (2.16). Once
each value of(fDk
)iis obtained, it becomes possible to construct the degree p poly-
nomial representation of fDk defined in equation (2.7). Upon transforming the flux in
equation (2.7) from the physical space to the reference space via equation (2.15), one
obtains the following expression for fD
fD =
Np∑
i=1
(fD)iℓi (x) . (2.26)
Figure (2.14) shows an example of fD for the case of p = 2.
ΩS
fD
Figure 2.14: An example of the approximate flux fD defined on the reference element ΩS
for the case of p = 2.
The sixth stage involves computing common values of the flux at the flux points
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 26
located on the edges of the reference element. The discontinuous flux in the reference
space (equation (2.26)) can be evaluated at the flux points on the left and right edges
in order to obtain fDL = fD (−1) and fD
R = fD (1). Figure (2.15) shows an example
of the approximate flux values fDL and fD
R for the case of p = 2.
fD
L
fD
R
ΩS
Figure 2.15: An example of the flux values fDL and fDR defined on the left and right edgesof the reference element ΩS for the case of p = 2.
Next, utilizing equation (2.15), fDL and fD
R can be transformed into the physical space,
where the resulting values are denoted by fDL− and fD
R−. A similar procedure can be
employed to compute the corresponding quantities fDL+ and fD
R+ in the neighboring
elements. Figure (2.16) shows an example of the approximate flux values fDL−, f
DR−,
fDL+, and f
DR+ for the case of p = 2.
fD
L−
fD
L+
fD
R+fD
R−
ΩkΩk−1 Ωk+1
Figure 2.16: An example of the flux values fDL−, fDR−, f
DL+, and f
DR+ defined on the left and
right edges of Ωk for the case of p = 2.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 27
Now, due to the different physical behaviors of the advective and diffusive fluxes, it
is convenient to divide fDL− , fD
R−, fDL+ , and fD
R+ into advective and diffusive parts,
such that fDL− = fD
L−, adv + fDL−, dif , f
DR− = fD
R−, adv + fDR−, dif , f
DL+ = fD
L+, adv + fDL+, dif ,
and fDR+ = fD
R+, adv + fDR+, dif .
Thereafter, a common value of the flux at the flux point on the left edge (denoted by
f ⋆L) can be computed by summing the common values of the advective and diffusive
fluxes on the left edge (denoted by f ⋆L, adv and f
⋆L, dif ), which are themselves computed
from fDL−, adv, f
DL+, adv, f
DL−, dif , and f
DL+, dif . Likewise, a common value of the flux at
the flux point on the right edge (denoted by f ⋆R) can be computed by summing the
common values of the advective and diffusive fluxes on the right edge (denoted by
f ⋆R, adv and f
⋆R, dif), which are themselves computed from fD
R−, adv, fDR+, adv, f
DR−, dif , and
fDR+, dif . Figure (2.17) shows an example of the common flux values f ⋆
L and f ⋆R for the
case of p = 2.
ΩkΩk−1 Ωk+1
f⋆
L
f⋆
R
Figure 2.17: An example of the flux values f⋆L and f⋆R defined on the left and right edges ofΩk for the case of p = 2.
Finally, in preparation for the next stage, the common flux values f ⋆L and f ⋆
R in the
physical space can be transformed to the reference space, yielding f ⋆L and f ⋆
R, in
accordance with equation (2.15). Figure (2.18) shows an example of the flux values
f ⋆L and f ⋆
R for the case of p = 2.
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 28
f⋆
R
f⋆
L
ΩS
Figure 2.18: An example of the flux values f⋆L and f⋆R defined on the left and right edges ofthe reference element ΩS for the case of p = 2.
The seventh stage involves forming the continuous flux in the reference space (f) by
adding a ‘flux correction’ (fC) to the discontinuous flux in the reference space (fD).
In order to ensure that f is continuous, the sum of fC and fD is required to equal
the common flux values f ⋆L and f ⋆
R at the flux points on the left and right edges of
the reference element, respectively. This requirement can be satisfied by defining fC
as follows
fC =(f ⋆L − fD
L
)hL (x) +
(f ⋆R − fD
R
)hR (x) , (2.27)
where hL (x) and hR (x) are correction functions that are required to have the same
properties as gL (x) and gR (x) from stage 3, as they must satisfy the following point-
wise constraints
hL (−1) = 1, hL (1) = 0, (2.28)
hR (−1) = 0, hR (1) = 1, (2.29)
and the following symmetry constraints
hL (x) = hR (−x) , hL (−x) = hR (x) . (2.30)
Figure (2.19) shows an example of fC for the case of p = 2
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 29
ΩS
fC
Figure 2.19: An example of fC defined on the reference element ΩS for the case of p = 2.
It should be noted that the correction functions hL (x) and hR (x) (used in the con-
struction of fC) are required to be of VCJH type (i.e. they are required to satisfy
equations (A.8) – (A.10)) in order to ensure that the resulting FR schemes are stable
for linear advection-diffusion problems in 1D [51]. For the sake of completeness, the
VCJH formulations of hL (x) and hR (x) are given in equations (A.11) and (A.12).
Now, having defined fC and the associated correction functions, one may construct
the continuous flux in the reference space (f) by summing fD and fC as follows
f = fD + fC = fD +(f ⋆L − fD
L
)hL (x) +
(f ⋆R − fD
R
)hR (x) . (2.31)
Figure (2.20) shows examples of f and fk for the case of p = 2, where fk =
f(Θk (x)
−1) in accordance with equation (2.15).
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 30
ΩS
f
(a) Example of f
ΩkΩk−1 Ωk+1
fk
fk−1
fk+1
(b) Example of fk
Figure 2.20: Examples of f and fk defined on the reference element ΩS and the physicalelement Ωk (respectively) for the case of p = 2.
The final stage involves computing the divergence of the continuous flux f and uti-
lizing it to update the discontinuous solution uD. Towards this end, consider substi-
tuting equations (2.26) and (2.31) into equation (2.17), and evaluating the result at
solution point j in order to obtain
∂(uD)j
∂t= −
[Np∑
i=1
(fD)i dℓi (x)
dx
] ∣∣∣∣x=xj
+
[(f ⋆L − fD
L
) dhL (x)
dx+(f ⋆R − fD
R
) dhR (x)
dx
] ∣∣∣∣x=xj
, (2.32)
where xj is the location of solution point j in reference space. Equation (2.32) can be
evaluated at all j = 1, · · · , Np solution points in order to obtain Np equations for the
CHAPTER 2. ESFR FOR ADV-DIFF PROBLEMS IN 1D 31
Np unknowns
[∂(uD)
1
∂t, . . . ,
∂(uD)Np
∂t
]in each element. Then, the resulting equations
can be marched forward in time using an explicit or implicit time-stepping scheme.
The behavior of the energy stable FR schemes in 1D is determined by the following
factors:
• The locations of the solution points xi.
• The methodology for computing the common solution values u⋆L and u⋆R.
• The methodology for computing the common flux values f ⋆L and f ⋆
R.
• The methodology for computing the energy stable FR solution correction func-
tions gL (and thus gR).
• The methodology for computing the energy stable FR flux correction functions
hL (and thus hR).
Vincent et al. [49] and Castonguay et al. [51] discuss how these factors effect the
energy stable FR schemes in 1D. In the following chapters, the 1D schemes will be
extended to higher dimensions (2D and 3D) and the stability of the resulting schemes
will be shown for linear advection-diffusion problems.
Chapter 3
Energy Stable Flux Reconstruction
for Advection-Diffusion Problems on
Triangles
32
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 33
3.1 Preamble
For the first time, this chapter constructs energy stable FR schemes (VCJH schemes)
for advection-diffusion problems on triangles, and proves the stability of these schemes
for linear advection-diffusion problems, for all orders of accuracy. Note that this chap-
ter is adapted from the article “Energy Stable Flux Reconstruction for Advection-
Diffusion Problems on Triangles” by D. M. Williams, P. Castonguay, P. E. Vin-
cent, and A. Jameson, which has been submitted to the Journal of Computational
Physics. The main results and over 95% of the text for this article were contributed
by D. M. Williams.
Due to the size of this chapter, it is useful to briefly examine its format. Section 2 of
this chapter presents a FR approach for solving advection-diffusion problems on trian-
gles, sections 3 and 4 define the VCJH correction fields on triangles, section 5 presents
plots of the VCJH correction fields, section 6 develops a range of VCJH schemes for
linear advection-diffusion problems on triangles, and proves that these schemes are
stable, and finally, section 7 presents results of linear numerical experiments involving
the newly proposed schemes.
3.2 Flux Reconstruction Approach for Advection-
Diffusion Problems on Triangles
In what follows, the FR schemes for advection problems on triangles, developed by
Castonguay et al. in [52], are extended to advection-diffusion problems. For readers
unfamiliar with the FR approach, the authors recommend a review of the 1D descrip-
tions of FR for advection in [23] and [49], or advection-diffusion in [51] and Chapter 2
(of this thesis) before proceeding further.
3.2.1 Preliminaries
Consider the 2D conservation law
∂u
∂t+∇·f(u,∇u) = 0, (3.1)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 34
where u is a scalar variable and f is a vector-valued flux. Equation (3.1) can be
rewritten as a first-order system,
∂u
∂t+∇·f(u,q) = 0, (3.2)
q−∇u = 0. (3.3)
The solution u = u(x, y, t) to the system evolves inside the 2D domainΩ (with bound-
ary Γ), where x and y are spatial coordinates, q = (qx, qy) is the auxiliary variable
with components qx = qx(u,∇u) and qy = qy(u,∇u), and f = (f, g) has components
f = f(u,q) and g = g(u,q). Assume that the domain Ω can be discretized into N
non-overlapping, conforming, straight-sided triangular elements Ωk such that
Ω =
N⋃
k=1
Ωk, (3.4)
Ωi ∩Ωj = ∅ ∀i 6= j. (3.5)
Approximations to equations (3.2) and (3.3) can be constructed within each element
Ωk. In equation (3.2), the exact solution u can be replaced by an approximate
solution uDk = uDk (x, y, t), which is a two-dimensional polynomial of degree p within
Ωk and is identically zero outside the element. In general, the sum of uDk and uDk+1 is
discontinuous at the interface between neighboring elements Ωk and Ωk+1, and as a
result each quantity is designated with a superscript D.
The flux f in equation (3.2) can be approximated by a function fk = (fk, gk) =
fk(x, y, t), which is a polynomial of degree p + 1 within Ωk and is identically zero
outside Ωk. The normal components of fk and fk+1 are required to be equivalent to
one another on the boundary between Ωk and Ωk+1.
Analogous approximations can be introduced into equation (3.3). Here, the auxiliary
variable q can be replaced by a function qDk = (qDx , q
Dy )k = qD
k (x, y, t) which is a
polynomial of degree p within Ωk and is identically zero outside. In general, the sum
of qDk and qD
k+1 is discontinuous at the interface between Ωk and Ωk+1. In addition,
the exact solution u can be replaced by an approximate solution uk = uk(x, y, t),
which is a polynomial of degree p + 1 within Ωk and is identically zero outside (and
where it is important to note that uk 6= uDk ). The approximate solutions uk and uk+1
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 35
are required to be equivalent to one another on the boundary between Ωk and Ωk+1.
Based on these definitions, the approximate first-order system within each element
Ωk becomes
∂uDk∂t
+∇·fk = 0, (3.6)
qDk −∇uk = 0. (3.7)
Next, in order to facilitate implementation, each element Ωk in physical space is
mapped to a reference right triangle ΩS. The mapping Θk for a linear triangular
element is
x = Θk(x) = −(x+ y)
2v1,k +
(x+ 1)
2v2,k +
(y + 1)
2v3,k, (3.8)
where x = (x, y) are the physical coordinates, x = (x, y) are the reference coordinates,
v1,k, v2,k, and v3,k are the vertex coordinates of the element Ωk in physical space,
and (−1,−1), (1,−1), and (−1, 1) are the vertex coordinates of the element ΩS in
reference space. Once a mapping is established, the physical quantities uk, uDk , fk, and
qDk can be reformulated as transformed quantities u, uD, f , and qD in the reference
domain using the following transformations due to Viviand [54] and Vinokur [55]
u = Jkuk(Θk(x), t), uD = JkuDk (Θk(x), t), (3.9)
f =
[fg
]= Jk J
−1k fk, qD =
[qDxqDy
]= ∇u = Jk J
Tk qD
k , (3.10)
where
fk =
[fkgk
], qD
k =
[qDxk
qDyk
]= ∇uk, (3.11)
and
Jk =
[∂x∂x
∂x∂y
∂y∂x
∂y∂y
], Jk = det(Jk). (3.12)
In equation (3.11), the operator ∇ is defined as the gradient in the reference domain
taken with respect to x and y (where thus far ∇ has been the gradient in the physical
domain taken with respect to x and y).
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 36
The following identities arise from equations (3.9)-(3.12)
∇·fk =1
Jk
(∇·f), (3.13)
∇uk · fk =1
J2k
(∇u · f
), (3.14)
∫
Ωk
Jkuk (∇·fk) dΩk =
∫
ΩS
u(∇·f)dΩS, (3.15)
∫
Ωk
Jk (∇uk · fk) dΩk =
∫
ΩS
∇u · fdΩS, (3.16)
∫
Γk
Jkuk (fk · n) dΓk =
∫
ΓS
u(f · n
)dΓS. (3.17)
These identities are used extensively in the proof of stability of the VCJH schemes
for the linear advection-diffusion problem on triangles (cf. section 3.6).
Upon utilizing equations (3.9)-(3.12) to reformulate the physical quantities in equa-
tions (3.6) and (3.7), one obtains
∂uD
∂t+ ∇·f = 0, (3.18)
qD − ∇u = 0. (3.19)
The next section will discuss the FR procedure for solving equations (3.18) and (3.19).
3.2.2 Procedure
The FR approach for advection-diffusion problems on triangles consists of seven
stages. In practice, when implementing the method, several of these stages can be
combined, however, in what follows they will be presented as separate stages for the
sake of clarity.
In the first stage of the FR approach on triangles, the approximate transformed
solution uD within the reference element ΩS is defined using a two-dimensional poly-
nomial of degree p. The polynomial is constructed from values of the approximate
transformed solution at Np =12(p+ 1)(p+ 2) solution points. (Figure (3.1) shows an
example of the locations of the solution points for the case of p = 2).
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 37
Figure 3.1: Example of solution point locations (denoted by circles) in the reference elementfor the case of p = 2.
The resulting approximate transformed solution takes the form
uD(x, t) =
Np∑
i=1
uDi ℓi(x), (3.20)
where uDi = Jk(xi) · uDk (Θk(xi), t) is the value of uD at solution point i, xi is the
location of solution point i, and ℓi(x) is the nodal basis function which takes on
the value of 1 at solution point i and the value of 0 at all other solution points.
The approximate transformed solution uD lies in the space Pp(ΩS), where Pp(ΩS)
defines the polynomial space of degree ≤ p on ΩS. Consequently, uD also lies in
the polynomial space Rp(ΓS) =φ|φ ∈ L2(ΓS), φ|Γf
∈ Pp(Γf), ∀Γf
which contains
polynomials of degree ≤ p defined on each side Γf of ΩS.
The second stage involves calculating common values for the transformed solution u⋆
at a set of flux points along the edges of ΩS. In what follows, a quantity labeled
with the indices f , j corresponds to a quantity at the flux point j of face f , where
1 ≤ f ≤ 3 and 1 ≤ j ≤ Nfp. Here, Nfp is the number of flux points on each face,
and is equal to p + 1. The location of flux point j on face f is denoted xf,j . The
convention used to number faces and flux points is illustrated in Figure (3.2).
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 38
j = 1 j = 2 j = Nfp
= 3
f = 1
f = 2
f = 3
Figure 3.2: Example of the numbering convention for the faces and flux points on thereference element for the case of p = 2. The flux points are denoted by squares and the fluxpoint index increases counterclockwise along an edge.
Common values of the transformed solution at each flux point (u⋆f,j) are computed
by first evaluating the multiply defined values of the physical solution uDf,j at each
flux point. At each flux point, uD(f,j)− is defined as the value of uDf,j computed using
the information local to the current element and uD(f,j)+ as the value of uDf,j computed
using information from the adjoining element. For a given element, the physical
solution at each flux point is obtained by evaluating the transformed solution (uD
in equation (3.20)) at each flux point, and converting the result to physical space
using equation (3.9). Once both physical solution values (uD(f,j)− and uD(f,j)+) are ob-
tained, an approach such as the Central Flux (CF) [13], Local Discontinuous Galerkin
where β is a directional parameter which allows u⋆f,j to assume a value that is biased
in either the upwind or downwind direction, and · and [[·]] are ‘average’ and ‘jump’
operators, respectively which are defined such that
uDf,j =uD(f,j)− + uD(f,j)+
2, [[uDf,j ]] = uD(f,j)−n− + uD(f,j)+n+, (3.22)
where n− and n+ denote the outward and inward pointing unit normals, respectively.
Note that choosing β = ±0.5n− promotes compactness of the FR scheme in multiple
dimensions, but does not ensure it for certain cases on general grids [13, 56]. Alter-
native approaches (such as CDG or BR2) can be employed to ensure compactness in
multiple dimensions. These compact approaches are essential for cases in which the
full matrix is to be formed, i.e. when a spatial discretization is paired with an im-
plicit time-stepping approach. However, this work is concerned with demonstrating
favorable aspects of pairing FR discretizations with explicit time-stepping approaches.
In this context, compactness is less important, and the LDG approach is frequently
sufficient.
After the common solution u⋆f,j is computed using the LDG approach or an alternative
approach, the transformed common solution denoted by u⋆f,j can be obtained from
equation (3.9).
Next, in preparation for the third stage, one may compute uCf,j the transformed solu-
tion correction at each flux point as follows
uCf,j = u⋆f,j − uDf,j. (3.23)
Note that the superscript ‘C’ on uCf,j stands for ‘correction.’
The third stage involves constructing the transformed auxiliary variable qD from the
transformed solution correction uCf,j at each flux point. Towards this end, one may
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 40
expand qD (in equation (3.19)) in order to obtain
qD = ∇u = ∇uD + ∇uC , (3.24)
where ∇uD is computed by applying the transformed gradient operator ∇ to uD in
equation (3.20), and ∇uC is computed by ‘lifting’ values of uCf,j from the element
boundary as follows
∇uC(x) =3∑
f=1
Nfp∑
j=1
uCf,j ψf,j(x) nf,j. (3.25)
The function ψf,j is a ‘correction field’ or ‘lifting operator’ which transforms uC defined
on ΓS into ∇uC defined throughout ΩS. Details regarding the construction of ψf,j
are given in section 3.4.
On rewriting equation (3.24) in terms of uD from equation (3.20) and ∇uC from
equation (3.25), one obtains the following expression for qD
qD(x) = ∇uD(x) + ∇uC(x) =Np∑
i=1
uDi ∇ℓi(x) +3∑
f=1
Nfp∑
j=1
uCf,j ψf,j(x) nf,j. (3.26)
In the fourth stage, the transformed discontinuous flux fD can be computed using
uD from equation (3.20) and qD from equation (3.26). Each component of the flux
fD = (fD, gD) can be expressed using a degree p polynomial as follows
fD(x) =
Np∑
i=1
fDi ℓi(x), gD(x) =
Np∑
i=1
gDi ℓi(x), (3.27)
where the quantities fDi and gDi are the components of the transformed flux at solution
point i, and are evaluated using uDi and qDi .
The fifth stage involves calculating transformed numerical fluxes f⋆ at flux points
along the edges of ΩS. At each flux point, the transformed flux f⋆f,j is calculated
based on the physical flux f⋆f,j which is composed from advective and diffusive parts
f⋆f,j = f⋆(f,j) adv+f⋆(f,j) dif . In turn, f⋆(f,j) adv and f⋆(f,j) dif are constructed from the multiply
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 41
defined discontinuous fluxes at each flux point denoted by
fD(f,j) adv− = f(uD(f,j)−
), fD(f,j) adv+ = f
(uD(f,j)+
), (3.28)
fD(f,j) dif− = f(uD(f,j)−,q
D(f,j)−
), fD(f,j) dif+ = f
(uD(f,j)+,q
D(f,j)+
). (3.29)
The solution values uD(f,j)− and uD(f,j)+ were obtained in stage 2, and the auxiliary vari-
able values qD(f,j)− and qD
(f,j)+ are obtained by evaluating qD at each flux point using
equation (3.26), and converting the result to physical space using equation (3.10).
Once fD(f,j) adv−, fD(f,j) adv+, f
D(f,j) dif−, and fD(f,j) dif+ are obtained, the numerical advec-
tive flux f⋆(f,j) adv and the numerical diffusive flux f⋆(f,j) dif can be computed. For linear
advection-diffusion equations, f⋆(f,j) adv can be computed using the Lax-Friedrichs ap-
proach (as defined in [13]), and for the nonlinear Navier-Stokes equations, f⋆(f,j) advcan be computed using a Roe [27] or Rusanov [60] approach. For both linear and
nonlinear equations, the numerical diffusive flux f⋆(f,j) dif can be computed using one
of the aforementioned CF, LDG, CDG, IP, BR1, or BR2 approaches. In particular,
if one elects to employ the LDG approach, the common numerical diffusive flux is
Note that the parameter β in equation (3.30) is preceded by a ‘+’ sign and in equa-
tion (3.21) it is preceded by a ‘−’ sign. Opposite signs help ensure symmetry of the
diffusive process in the following sense: if the common solution u⋆f,j is upwind biased
then the numerical flux f⋆(f,j) dif is downwind biased or vice versa.
Once the total numerical flux f⋆f,j = f⋆(f,j) adv + f⋆(f,j) dif is computed, the transformed
numerical flux denoted by f⋆f,j can be obtained from equation (3.10).
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 42
The sixth stage involves forming the transformed correction flux fC which will be
added to the transformed discontinuous flux fD to form the transformed continuous
flux f (i.e. f = fD + fC). fC is constructed using vector correction functions hf,j (x)
which are restricted to lie in the Raviart-Thomas space [61] of order p, denoted by
RTp(ΩS). Note that RTp(ΩS) = (Pp (ΩS))2 + xPp (ΩS), where (Pp (ΩS))
2 is the two-
dimensional vector space with components with degree ≤ p. As a consequence, the
vector correction functions possess the following attributes
φf,j(x) ∈ Pp(ΩS), hf,j(x) · n|ΓS∈ Rp(ΓS), (3.33)
where
φf,j(x) ≡ ∇·hf,j(x) (3.34)
is referred to as the ‘correction field’, and hf,j(x) · n|ΓSis referred to as the ‘normal
trace.’
Furthermore, if k represents a particular face (1 ≤ k ≤ 3) and ℓ represents a particular
flux point (1 ≤ ℓ ≤ Nfp), each of the correction functions hf,j is required to satisfy
hf,j(xk,ℓ) · nk,ℓ =
1 if f = k and j = ℓ0 if f 6= k or j 6= ℓ
. (3.35)
Equations (3.33) and (3.35) ensure that the resulting FR schemes are conservative
[52]. Figure (3.3) shows an example of a vector correction function hf,j which satisfies
equations (3.33) and (3.35) for the case of p = 2.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 43
Figure 3.3: Example of a vector correction function hf,j associated with flux pointf = 2, j = 2 for the case of p = 2.
Now, having defined the vector correction functions hf,j , one may define the trans-
formed correction flux fC as follows
fC(x) =
3∑
f=1
Nfp∑
j=1
[(f⋆f,j − fDf,j
)· nf,j
]hf,j(x) =
3∑
f=1
Nfp∑
j=1
∆f,jhf,j(x). (3.36)
Here, ∆f,j is defined as the difference between the normal transformed numerical flux
and the normal transformed discontinuous flux at the flux point f ,j.
The final stage involves calculating the divergence of the continuous transformed flux
f . The result is used to update the transformed solution at the solution point xi as
follows
duDidt
= −(∇·f) ∣∣∣∣
xi
= −(∇·fD
) ∣∣∣∣xi
−(∇·fC
) ∣∣∣∣xi
= −Np∑
n=1
(fDn
) ∂ℓn∂x
∣∣∣∣xi
−Np∑
n=1
(gDn) ∂ℓn∂y
∣∣∣∣xi
−3∑
f=1
Nfp∑
j=1
∆f,jφf,j(xi). (3.37)
For triangles, the overall behavior of the FR scheme depends on six factors, namely:
1. The location of the solution points xi.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 44
2. The location of the flux points xf,j .
3. The methodology for calculating the transformed common solution values u⋆f,j.
4. The methodology for calculating the transformed numerical flux values f⋆f,j.
5. The form of the solution correction field ψf,j .
6. The form of the flux correction field φf,j.
The FR schemes can be simplified by choosing the same solution and flux correction
fields (choosing ψf,j = φf,j). Nevertheless, in an effort to generalize the subsequent
discussions, the solution and flux correction fields are assumed to be distinct unless
otherwise indicated.
Finally, note that the computational effort associated with the FR approach is iden-
tical to that associated with a collocation-based nodal DG approach. When the cor-
rections fields are chosen appropriately, the FR approach recovers a collocation-based
nodal DG scheme (as mentioned previously).
3.3 VCJH Flux Correction Fields on Triangles
For linear advection problems on triangles, Castonguay et al. [52] have identified a
range of correction fields φf,j for correcting the flux, which lead to energy stable FR
schemes. The correction fields φf,j are polynomials of degree p, and can therefore be
expressed in terms of an orthonormal basis Ln of degree p as follows
φf,j =
Np∑
n=1
σnLn(x), (3.38)
where the coefficients σn are constants. Here each Ln is a member of the 2D Dubiner
basis [62] defined on the reference right triangle ΩS. This basis is given by
Ln(x) =√2Qv(a)Q
(2v+1,0)w (b)(1− b)v, (3.39)
k = w + (p + 1)v + 1− v
2(v − 1), (v, w) ≥ 0, v + w ≤ p,
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 45
where
a = 2(1 + x)
(1− y) − 1, b = y, (3.40)
and Q(α,β)m is the mth order Jacobi polynomial (normalized as described in [13]).
Castonguay et al. [52] showed that stability for linear advection problems can be
achieved if the coefficients σn are computed by solving the following system of equa-
tions
Np∑
n=1
σn
p+1∑
v=1
cv
(D(p,v)Li
)(D(p,v)Ln
)= −σi +
∫
ΓS
(hf,j · n)Li dΓS for 1 ≤ i ≤ Np,
(3.41)
where each dot product (hf,j · n) is defined via equation (3.35), each derivative oper-
ator D(p,v) is of the form
D(p,v) =∂p
∂x(p−v+1)∂y(v−1), 1 ≤ v ≤ p+ 1, (3.42)
and each constant cv is of the form
cv = c
(p
v − 1
)= c
[p!
(v − 1)!(p− v + 1)!
]. (3.43)
In equation (3.43), c is a scalar, where 0 ≤ c ≤ ∞. The correction fields φf,j obtained
from the solution of equation (3.41) lead to VCJH schemes on triangles which are an
infinite family of FR schemes parameterized by c. For linear advection problems on
triangles, if c = cdg = 0, a collocation-based nodal DG scheme is recovered [52].
The stability of VCJH schemes for linear advection problems on triangles (for which
periodic boundary conditions can be imposed) is ensured because it can be shown
that a Sobolev-type norm of the solution is non-increasing, i.e.
d
dt‖UD‖2p,c ≤ 0, (3.44)
where the Sobolev-type norm is defined as
‖UD‖p,c =
N∑
k=1
∫
Ωk
[(uDk )
2 +1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
1/2
, (3.45)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 46
UD is the total (domain-wide) solution defined as
UD =
N∑
k=1
uDk , (3.46)
and AS denotes the area of the reference element ΩS. Note that, in equation (3.45),
the derivative operator D(p,v) in ΩS has been applied to uDk in the physical domain
Ωk using the chain rule.
3.4 VCJH Solution Correction Fields on Triangles
Consider the solution correction function gf,j and solution correction field (‘lifting
operator’) ψf,j , where ψf,j = ∇·gf,j. According to the previous section, the correction
field ψf,j can be classified as a ‘VCJH correction field’ if it takes the following form
ψf,j =
Np∑
n=1
ξnLn(x), (3.47)
where each coefficient ξn is obtained from the following system of equations
Np∑
n=1
ξn
p+1∑
v=1
κv
(D(p,v)Li
)(D(p,v)Ln
)= −ξi +
∫
ΓS
(gf,j · n)Li dΓS for 1 ≤ i ≤ Np,
(3.48)
where each D(p,v) is a derivative operator of degree p (defined by equation (3.42) in
section 3.3).
In equation (3.48), each constant κv is defined as
κv = κ
(p
v − 1
)= κ
[p!
(v − 1)!(p− v + 1)!
], 0 ≤ κ ≤ ∞, (3.49)
and hence the correction fields ψf,j are parameterized by the scalar κ.
As currently constructed, the correction fields ψf,j possess an important ability to act
as ‘lifting operators’, which transform uC defined on the element boundary ΓS into
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 47
∇uC defined within the element interior ΩS, according to the following identity
∫
ΩS
∇uC dΩS =
∫
ΓS
uCn dΓS. (3.50)
In what follows, the derivation of this identity will be examined.
First, recall that ψf,j = ∇ · gf,j, and thus according to the divergence theorem, gf,j
and ψf,j are related as follows
∫
ΩS
ψf,j dΩS =
∫
ΓS
gf,j · n dΓS. (3.51)
The quantity gf,j ·n (which satisfies equation (3.35) with gf,j in place of hf,j) vanishes
on all faces except for face f . As a result, equation (3.51) becomes
∫
ΩS
ψf,j dΩS =
∫
Γf
gf,j · nf,j dΓf . (3.52)
Multiplying equation (3.52) by uCf,j and nf,j , and summing over f and j yields
∫
ΩS
3∑
f=1
Nfp∑
j=1
uCf,jψf,jnf,j dΩS =
3∑
f=1
∫
Γf
Nfp∑
j=1
uCf,j (gf,j · nf,j) nf,j dΓf
. (3.53)
Substituting equation (3.25) into this equation, and replacing the term gf,j · nf,j with
the 1D Lagrange polynomial ℓ 1Df,j (because gf,j · nf,j satisfies equation (3.35) and
gf,j ∈ RTp(ΩS)), one obtains
∫
ΩS
∇uC dΩS =
3∑
f=1
∫
Γf
Nfp∑
j=1
uCf,j ℓ1Df,j nf,j dΓf
. (3.54)
Note that the 1D Lagrange polynomial ℓ 1Df,j is a degree p polynomial on face f , which
takes on the value of 1 at flux point j, and the value of 0 at all other flux points on
the face.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 48
A corollary to the divergence theorem requires that
∫
ΩS
∇uC dΩS =
∫
ΓS
uCn dΓS =3∑
f=1
[∫
Γf
uC∣∣Γf
nf dΓf
]. (3.55)
From equations (3.54) and (3.55), one obtains
3∑
f=1
[∫
Γf
uC∣∣Γf
nf dΓf
]=
3∑
f=1
∫
Γf
Nfp∑
j=1
uCf,j ℓ1Df,j nf,j dΓf
. (3.56)
Each domain Γf is arbitrary (as the standard element and its associated faces can
be chosen arbitrarily) and thus the integrands in equation (3.56) must be equal.
Consequently, uC on ΓS takes the form
uC∣∣∣∣Γf
=
Nfp∑
j=1
uCf,j ℓ1Df,j uC
∣∣∣∣ΓS
=
uC∣∣Γ1
on Γ1
uC∣∣Γ2
on Γ2
uC∣∣Γ3
on Γ3
. (3.57)
In conclusion, the VCJH correction fields ψf,j serve as ‘lifting operators’, because
they provide a relationship (given by equation (3.55)) between uC defined on ΓS
(equation (3.57)) and ∇uC defined within ΩS (equation (3.25)).
3.5 Visualization of VCJH Correction Fields
The VCJH flux and solution correction fields φf,j and ψf,j assume the same analytical
form (with the caveat that the parameterizing coefficients c and κ need not be the
same), and therefore, for the sake of brevity this section will only present plots of
the flux correction fields φf,j. In the following figures, contours of the fields φf,j are
plotted for the case of p = 2 and c = c+ = 3.13 × 10−2. It should be noted that the
scheme with c = c+ is optimal in the following sense: for linear advection problems,
the combination of the VCJH scheme with the value of c = c+ and the 5-stage, 4th
order, Runge-Kutta time-stepping scheme (derived in [16]) yielded a maximal explicit
time-step limit relative to all other schemes tested by Castonguay et al. in [52].
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 49
f=1, j=1 f=1, j=2
f=1, j=3
Figure 3.4: Plots of the VCJH correction fields φf,j for the case of p = 2, f = 1, andc = c+ = 3.13 × 10−2.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 50
f=2, j=1 f=2, j=2
f=2, j=3
Figure 3.5: Plots of the VCJH correction fields φf,j for the case of p = 2, f = 2, andc = c+ = 3.13 × 10−2.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 51
f=3, j=1 f=3, j=2
f=3, j=3
Figure 3.6: Plots of the VCJH correction fields φf,j for the case of p = 2, f = 3, andc = c+ = 3.13 × 10−2.
From Figures (3.4) – (3.6), it is clear that the VCJH correction fields behave in accor-
dance with physical intuition, as the symmetries of the correction fields are consistent
with the underlying symmetries of the right triangle. This is expected, as Castonguay
et al. [52] demonstrated (via numerical experiments) that if the VCJH correction fields
are parameterized according to equations (3.43) and (3.49), the resulting fields are
symmetric under rotations and reflections of the triangle unto itself.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 52
3.6 Proof of Stability of VCJH Schemes for the
Linear Advection-Diffusion Equation on Tri-
angles
In this section, it is shown that if the correction fields ψf,j and φf,j are chosen to be
VCJH correction fields, (i.e. ψf,j = ∇ ·gf,j and φf,j = ∇ ·hf,j where hf,j and gf,j are
VCJH vector correction functions), and the transformed common solution values u⋆f,jand numerical flux values f⋆f,j are obtained appropriately, the resulting FR schemes
are stable for the linear advection-diffusion equation on straight-sided triangles.
3.6.1 Preliminaries
The linear advection-diffusion equation in 2D can be expressed as the following first-
order system
∂u
∂t+∇· (au− bq) = 0, (3.58)
q−∇u = 0, (3.59)
where a = (ax, ay) is a velocity vector (with constant components) and b is a dif-
fusivity coefficient (with constant value b ≥ 0). Equations (3.58) and (3.59) can be
rewritten in terms of physical coordinates in the kth element (Ωk), yielding equa-
tions (3.6) and (3.7), where fk = fadv,k + fdif,k, and where fadv,k = auDk and fdif,k =
−bqDk .
3.6.2 The Stability of VCJH Schemes
The stability of VCJH schemes for linear advection-diffusion problems on triangles
can be established by examining the evolution in time of the Sobolev-type norm of
the solution given in equation (3.45). An expression for the time rate of change of
this norm will be constructed using contributions from the time rate of change of the
solution itself∂uD
k
∂t(equation (3.6)) and the auxiliary variable qD
k (equation (3.7)).
The following lemmas will manipulate and combine equations (3.6) and (3.7) in order
to establish an upper bound for the time rate of change of the norm.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 53
Lemma 3.6.1. For all FR schemes, equation (3.6) holds, and therefore the following
results also hold
Jk2
d
dt
∫
Ωk
(uDk)2dΩk = −
∫
ΩS
uD(∇·fD + ∇·fC
)dΩS (3.60)
and
Jk2AS
d
dt
∫
Ωk
(D(p,v) uDk
)2dΩk = −
(D(p,v) uD
)(D(p,v)
(∇·fC
)). (3.61)
Proof. The following relation holds for all FR schemes: ∇·fk = ∇·fDk +∇·fCk . Upon
combining this relation and equation (3.6) and rearranging the result, one obtains
∂uDk∂t
= −∇·fDk −∇·fCk . (3.62)
Multiplying this equation by JkuDk and integrating over the element domain Ωk yields
the following
Jk2
d
dt
∫
Ωk
(uDk)2dΩk = −
∫
Ωk
JkuDk
(∇·fDk
)dΩk −
∫
Ωk
JkuDk
(∇·fCk
)dΩk. (3.63)
Upon replacing the integrals over the physical domain Ωk on the RHS of equa-
tion (3.63) with integrals over the reference domain ΩS (using the identity from
equation (3.15)), one obtains equation (3.60), the first result of Lemma 3.6.1.
The second result of Lemma 3.6.1 can be derived as follows. Consider applying the
operator D(p,v) to both sides of equation (3.62) in order to obtain
∂
∂t
(D(p,v) uDk
)= −D(p,v)
(∇·fDk
)− D(p,v)
(∇·fCk
)= −D(p,v)
(∇·fCk
). (3.64)
Note that D(p,v)(∇·fDk
)= 0 because ∇·fDk is at most degree (p− 1).
If both sides of equation (3.64) are multiplied by (D(p,v) uDk ) and integrated over Ωk,
one obtains
1
2
d
dt
∫
Ωk
(D(p,v) uDk
)2dΩk = −
∫
Ωk
(D(p,v) uDk
)(D(p,v)
(∇·fCk
))dΩk. (3.65)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 54
Substituting equations (3.9) and (3.13) into equation (3.65) yields
1
2
d
dt
∫
Ωk
(D(p,v) uDk
)2dΩk = −
∫
ΩS
1
Jk
(D(p,v) uD
)(D(p,v)
(∇·fC
))dΩS
= −AS
Jk
(D(p,v) uD
)(D(p,v)
(∇·fC
)). (3.66)
Upon rearranging equation (3.66) one obtains equation (3.61). This completes the
proof of Lemma 3.6.1.
Lemma 3.6.2. For all FR schemes, equation (3.7) holds, and therefore the following
results also hold
Jk
∫
Ωk
fDdif,k · qDk dΩk =
∫
ΩS
fDdif ·(∇uD + ∇uC
)dΩS (3.67)
and
JkAS
∫
Ωk
(D(p,v) fDdif,k
)·(D(p,v) qD
k
)dΩk =
(D(p,v) fDdif
)·(D(p,v)
(∇uC
)). (3.68)
Proof. The following relation holds for all FR schemes: ∇uk = ∇uDk + ∇uCk . Upon
combining this relation and equation (3.7) and rearranging the result, one obtains
qDk = ∇uDk +∇uCk . (3.69)
Taking the dot product of equation (3.69) with Jk fDdif,k and integrating over the
element domain Ωk yields
Jk
∫
Ωk
fDdif,k · qDk dΩk = Jk
∫
Ωk
(fDdif,k · ∇uDk
)dΩk + Jk
∫
Ωk
(fDdif,k · ∇uCk
)dΩk. (3.70)
Upon transforming the RHS of equation (3.70) to the reference domain using equa-
tion (3.16) (with fdif,k and fdif in place of fk and f), one obtains equation (3.67), the
first result of Lemma 3.6.2.
The second result of Lemma 3.6.2 can be derived as follows. Consider applying the
operator D(p,v) to both sides of equation (3.69) in order to obtain
D(p,v) qDk = D(p,v)
(∇uDk
)+ D(p,v)
(∇uCk
)= D(p,v)
(∇uCk
). (3.71)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 55
The term D(p,v)(∇uDk
)vanishes because ∇uDk is of degree (p − 1). Taking the dot
product of equation (3.71) with(D(p,v) fDdif,k
)and integrating over Ωk yields
∫
Ωk
(D(p,v) fDdif,k
)·(D(p,v) qD
k
)dΩk =
∫
Ωk
(D(p,v) fDdif,k
)·(D(p,v)
(∇uCk
))dΩk.
(3.72)
The RHS of equation (3.72) can be simplified via the identities in equations (3.10)
and (3.11). On substituting fDdif,k, fDdif , u
Ck , and uC in place of fk, f , uk, and u in
equations (3.10) and (3.11), and rearranging the result, one obtains
fDdifJk
= J−1k fDdif,k, (3.73)
∇uCJk
= JTk ∇uCk . (3.74)
Upon applying the derivative operator D(p,v) to equations (3.73) and (3.74), one
obtains
1
JkD(p,v)
(fDdif
)= D(p,v)
(J−1k fDdif,k
), (3.75)
1
JkD(p,v)
(∇uC
)= D(p,v)
(JTk ∇uCk
). (3.76)
Next, consider taking the dot product of equations (3.75) and (3.76) as follows
D(p,v)(J−1k fDdif,k
)· D(p,v)
(JTk ∇uCk
)=
1
J2k
(D(p,v) fDdif
)·(D(p,v)
(∇uC
)). (3.77)
Now, one should note that, for straight-sided triangles Jk is a matrix of constant
values (independent of x and y) and therefore the following identities hold
D(p,v)(J−1k fDdif,k
)= J−1
k D(p,v)(fDdif,k
), (3.78)
D(p,v)(JTk ∇uCk
)= JT
k D(p,v)
(∇uCk
). (3.79)
On substituting equations (3.78) and (3.79) into equation (3.77) and simplifying the
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 56
result, one obtains
(D(p,v) fDdif,k
)·(D(p,v)
(∇uCk
))=
1
J2k
(D(p,v) fDdif
)·(D(p,v)
(∇uC
)). (3.80)
Upon substituting equation (3.80) into the RHS of equation (3.72), one obtains
∫
Ωk
(D(p,v) fDdif,k
)·(D(p,v) qD
k
)dΩk =
∫
Ωk
1
J2k
(D(p,v) fDdif
)·(D(p,v)
(∇uC
))dΩk
=AS
Jk
(D(p,v) fDdif
)·(D(p,v)
(∇uC
)). (3.81)
On rearranging equation (3.81) one obtains equation (3.68). This completes the proof
of Lemma 3.6.2.
Lemma 3.6.3. If φf,j and ψf,j are the VCJH correction fields, the following identity
holds
Jk2
d
dt
∫
Ωk
(uDk)2dΩk +
Jk2AS
d
dt
∫
Ωk
p+1∑
v=1
cv
(D(p,v) uDk
)2dΩk
− Jk∫
Ωk
fDdif,k · qDk dΩk −
JkAS
∫
Ωk
p+1∑
v=1
κv
(D(p,v) fDdif,k
)·(D(p,v) qD
k
)dΩk
= −∫
ΩS
uD(∇·fD
)dΩS −
∫
ΓS
uD(fC · n
)dΓS
−∫
ΩS
fDdif ·(∇uD
)dΩS −
∫
ΓS
uC(fDdif · n
)dΓS, (3.82)
where constants cv and κv are related via equations (3.43) and (3.49) to constants
c and κ, respectively, which parameterize the VCJH correction fields φf,j and ψf,j,
respectively.
Proof. Consider subtracting equation (3.67) from equation (3.60) in order to obtain
Jk2
d
dt
∫
Ωk
(uDk)2dΩk − Jk
∫
Ωk
fDdif,k · qDk dΩk
= −∫
ΩS
uD(∇·fD + ∇·fC
)dΩS −
∫
ΩS
fDdif ·(∇uD + ∇uC
)dΩS. (3.83)
Next, consider multiplying equation (3.61) by cv and equation (3.68) by−κv, summing
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 57
each equation over v, and adding the results to equation (3.83) in order to obtain
Jk2
d
dt
∫
Ωk
(uDk)2dΩk +
Jk2AS
d
dt
∫
Ωk
p+1∑
v=1
cv
(D(p,v) uDk
)2dΩk
− Jk∫
Ωk
fDdif,k · qDk dΩk −
JkAS
∫
Ωk
p+1∑
v=1
κv
(D(p,v) fDdif,k
)·(D(p,v) qD
k
)dΩk
= −∫
ΩS
uD(∇·fD + ∇·fC
)dΩS −
p+1∑
v=1
cv
(D(p,v) uD
)(D(p,v)
(∇·fC
))
−∫
ΩS
fDdif ·(∇uD + ∇uC
)dΩS −
p+1∑
v=1
κv
(D(p,v) fDdif
)·(D(p,v)
(∇uC
)). (3.84)
Suppose that ∇·fC and ∇uC are constructed based on correction fields φf,j = ∇ ·hf,j
and ψf,j = ∇·gf,j, respectively, which satisfy the following identities (that are satisfied
if hf,j and gf,j are the VCJH correction functions as shown in [52])
∫
ΩS
hf,j · ∇Li dΩS −p+1∑
v=1
cv
(D(p,v) Li
)(D(p,v) φf,j
)= 0, (3.85)
∫
ΩS
gf,j · ∇Li dΩS −p+1∑
v=1
κv
(D(p,v) Li
)(D(p,v) ψf,j
)= 0. (3.86)
One may manipulate equations (3.85) and (3.86) in order to obtain two additional
equations that can be utilized to simplify equation (3.84). Towards this end, one may
perform integration by parts on equation (3.85) and rearrange the result, in order to
obtain
∫
ΓS
(hf,j · n)Li dΓS =
∫
ΩS
φf,jLi dΩS +
p+1∑
v=1
cv
(D(p,v)Li
)(D(p,v)φf,j
). (3.87)
Note that uD ∈ Pp (ΩS) can be written as a linear combination of the orthonormal
polynomials Li. As a result, equation (3.87) can be rewritten in terms of uD as follows
∫
ΓS
(hf,j · n) uD dΓS =
∫
ΩS
φf,j uD dΩS +
p+1∑
v=1
cv
(D(p,v)uD
)(D(p,v)φf,j
). (3.88)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 58
On multiplying equation (3.88) by ∆f,j and summing over f and j, one obtains
∫
ΓS
3∑
f=1
Nfp∑
j=1
∆f,j hf,j · n
uD dΓS
=
∫
ΩS
3∑
f=1
Nfp∑
j=1
∆f,j φf,j
uD dΩS +
p+1∑
v=1
cv
(D(p,v)uD
)D(p,v)
3∑
f=1
Nfp∑
j=1
∆f,j φf,j
.
(3.89)
Upon substituting the definition of fC (equation (3.36)) into equation (3.89) and
rearranging the result, one obtains
p+1∑
v=1
cv
(D(p,v) uD
)(D(p,v)
(∇·fC
))=
∫
ΓS
uD(fC · n
)dΓS −
∫
ΩS
uD(∇ · fC
)dΩS.
(3.90)
Setting equation (3.90) aside for the moment, consider performing integration by
parts on equation (3.86) and rearranging the result, in order to obtain
∫
ΓS
(gf,j · n)Li dΓS =
∫
ΩS
ψf,jLi dΩS +
p+1∑
v=1
κv
(D(p,v)Li
)(D(p,v)ψf,j
). (3.91)
On the LHS, gf,j · n vanishes on all faces except for face f (as required by equa-
tion (3.35)). As a result, equation (3.91) becomes
∫
Γf
(gf,j · nf,j)Li dΓf =
∫
ΩS
ψf,jLi dΩS +
p+1∑
v=1
κv
(D(p,v)Li
)(D(p,v)ψf,j
). (3.92)
Because (fDdif · nf,j) ∈ Pp (ΩS), it can be expressed as a linear combination of the
orthonormal polynomials Li and therefore, equation (3.92) can be rewritten with
(fDdif · nf,j) in place of Li as follows
∫
Γf
(gf,j · nf,j)(fDdif · nf,j
)dΓf
=
∫
ΩS
ψf,j
(fDdif · nf,j
)dΩS +
p+1∑
v=1
κv
(D(p,v)
(fDdif · nf,j
))(D(p,v)ψf,j
). (3.93)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 59
On multiplying equation (3.93) by uCf,j and summing over f and j, one obtains
3∑
f=1
∫
Γf
Nfp∑
j=1
uCf,j (gf,j · nf,j)(fDdif · nf,j
)dΓf
=
∫
ΩS
fDdif ·
3∑
f=1
Nfp∑
j=1
uCf,j ψf,jnf,j
dΩS +
p+1∑
v=1
κv D(p,v)(fDdif ) · D(p,v)
3∑
f=1
Nfp∑
j=1
uCf,j ψf,jnf,j
.
(3.94)
Upon combining equations (3.25), (3.57), and (3.94) and rearranging the result, one
obtains
p+1∑
v=1
κv
(D(p,v) fDdif
)·(D(p,v)
(∇uC
))=
∫
ΓS
uC(fDdif · n
)dΓS−
∫
ΩS
(fDdif · ∇uC
)dΩS.
(3.95)
Upon substituting equations (3.90) and (3.95) into the RHS of equation (3.84), one
obtains equation (3.82). This completes the proof of Lemma 3.6.3.
Theorem 3.6.1. If the VCJH schemes on triangles (for which Lemmas 3.6.1–3.6.3
hold) are employed in conjunction with the Lax-Friedrichs formulation for the advec-
tive numerical flux f⋆adv
f⋆adv = fDadv+λ
2
(max
u∈[uD−,uD
+ ]
∣∣∣∣∂fadv∂u· n∣∣∣∣
)[[uD]], (3.96)
with 0 ≤ λ ≤ 1, and the LDG formulation for the common solution u⋆ and diffusive
numerical flux f⋆dif ,
u⋆ = uD − β · [[uD]], (3.97)
f⋆dif = fDdif+ τ [[uD]] + β[[fDdif ]], (3.98)
with β = (βx, βy) and τ ≥ 0, then it can be shown that the following result holds
d
dt‖UD‖2p,c ≤ 0. (3.99)
Proof. The fluxes fC and fD in equation (3.82) (from Lemma 3.6.3) can be ex-
pressed in terms of advective and diffusive parts: in particular fC = fCadv + fCdif and
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 60
fD = fDadv + fDdif . In addition, the diffusive flux fdif,k on the LHS of equation (3.82)
is defined such that fdif,k = −bqDk (as mentioned previously). As a result, equa-
tion (3.82) becomes
Jk2
d
dt
∫
Ωk
[(uDk)2
+1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
+ b Jk
∫
Ωk
[qDk · qD
k +1
AS
p+1∑
v=1
κv
(D(p,v) qD
k
)·(D(p,v) qD
k
)]dΩk
= −∫
ΩS
uD(∇·fDadv
)dΩS −
∫
ΓS
uD(fCadv · n
)dΓS −
∫
ΩS
uD(∇·fDdif
)dΩS
−∫
ΩS
(∇uD · fDdif
)dΩS −
∫
ΓS
uD(fCdif · n
)dΓS −
∫
ΓS
uC(fDdif · n
)dΓS. (3.100)
Setting equation (3.100) aside for the moment, one may consider the following identity
uD(∇·fDadv) =1
2∇ · (uD fDadv), (3.101)
which holds because fDadv = J−1k auD, J−1
k is a constant matrix, and a is a constant
vector. On substituting equation (3.101) into the RHS of equation (3.100), and
employing the divergence theorem and integration by parts, one obtains
Jk2
d
dt
∫
Ωk
[(uDk)2
+1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
+ b
∫
Ωk
[qDk · qD
k +1
AS
p+1∑
v=1
κv
(D(p,v) qD
k
)·(D(p,v) qD
k
)]dΩk
= −12
∫
ΓS
uD(fDadv · n
)dΓS −
∫
ΓS
uD(fCadv · n
)dΓS
−∫
ΓS
uD(fDdif · n
)dΓS −
∫
ΓS
uD(fCdif · n
)dΓS −
∫
ΓS
uC(fDdif · n
)dΓS. (3.102)
Consider the following identities
u
∣∣∣∣ΓS
=(uD + uC
) ∣∣∣∣ΓS
= u⋆, f · n∣∣∣∣ΓS
=(fD + fC
)· n∣∣∣∣ΓS
= f⋆ · n, (3.103)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 61
where (on each face Γf)
u⋆ =
Nfp∑
j=1
u⋆f,j ℓ1Df,j , f⋆ =
Nfp∑
j=1
f⋆f,j ℓ1Df,j . (3.104)
One may use equation (3.103) to eliminate fC and uC from equation (3.102), and
thereafter one may use equation (3.17) to transform the integrals on the RHS in
order to obtain the following expression
1
2
d
dt
∫
Ωk
[(uDk)2
+1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
+ b
∫
Ωk
[qDk · qD
k +1
AS
p+1∑
v=1
κv
(D(p,v) qD
k
)·(D(p,v) qD
k
)]dΩk
=
∫
Γk
[1
2uDk(fDadv,k · n
)− uDk (f⋆adv · n)
]dΓk
+
∫
Γk
[uDk(fDdif,k · n
)− uDk
(f⋆dif · n
)− u⋆
(fDdif,k · n
)]dΓk, (3.105)
where the quantities u⋆, f⋆adv, and f⋆dif are defined (on each face Γf) as
u⋆ =
Nfp∑
j=1
u⋆f,j ℓ1Df,j , f⋆adv =
Nfp∑
j=1
f⋆(f,j) adv ℓ1Df,j , f⋆dif =
Nfp∑
j=1
f⋆(f,j) dif ℓ1Df,j . (3.106)
To obtain a description of the solution behavior within the entire domain, one must
sum over all the elements in the mesh (summing over k on equation (3.105)) as follows
1
2
d
dt
N∑
k=1
∫
Ωk
[(uDk)2
+1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
+ bN∑
k=1
∫
Ωk
[qDk · qD
k +1
AS
p+1∑
v=1
κv
(D(p,v) qD
k
)·(D(p,v) qD
k
)]dΩk
=
N∑
k=1
∫
Γk
[1
2uDk(fDadv,k · n
)− uDk (f⋆adv · n)
]dΓk
+N∑
k=1
∫
Γk
[uDk(fDdif,k · n
)− uDk
(f⋆dif · n
)− u⋆
(fDdif,k · n
)]dΓk
. (3.107)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 62
For constants cv and κv that satisfy 0 ≤ cv <∞ and 0 ≤ κv <∞, the expressions
‖UD‖p,c =
N∑
k=1
∫
Ωk
[(uDk)2
+1
AS
p+1∑
v=1
cv
(D(p,v) uDk
)2]dΩk
1/2
(3.108)
and
‖QD‖p,κ =
N∑
k=1
∫
Ωk
[qDk · qD
k +1
AS
p+1∑
v=1
κv
(D(p,v) qD
k
)·(D(p,v) qD
k
)]dΩk
1/2
(3.109)
are broken Sobolev-type norms of the solution UD and the auxiliary variable QD,
respectively. The expressions
Θadv =
N∑
k=1
∫
Γk
[1
2uDk(fDadv,k · n
)− uDk (f⋆adv · n)
]dΓk
(3.110)
and
Θdif =N∑
k=1
∫
Γk
[uDk(fDdif,k · n
)− uDk
(f⋆dif · n
)− u⋆
(fDdif,k · n
)]dΓk
(3.111)
represent contributions from the advective and diffusive fluxes at the element bound-
aries. Using equations (3.108) - (3.111), equation (3.107) can be rewritten as
1
2
d
dt‖UD‖2p,c = −b ‖QD‖2p,κ +Θadv +Θdif . (3.112)
On the RHS of equation (3.112), the term −b ‖QD‖2p,κ is guaranteed to be non-
positive for b ≥ 0, and the terms Θadv and Θdif are guaranteed to be non-positive
for appropriate choices of the common solution u⋆, and the common numerical fluxes
f⋆adv and f⋆dif .
The Lax-Friedrichs formulation provides a suitable expression for f⋆adv (equation (3.96)).
Upon substituting fadv = au into equation (3.96) and taking the dot product with n,
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 63
one obtains
f⋆adv · n =
(auD+ λ
2|a · n| [[uD]]
)· n
=1
2
(uD+ + uD−
)(axnx + ayny) +
λ
2|axnx + ayny|
(uD− − uD+
), (3.113)
where (on each face Γf)
uD+ =
Nfp∑
j=1
uD(f,j)+ ℓ1Df,j , uD− =
Nfp∑
j=1
uD(f,j)− ℓ1Df,j , (3.114)
and where λ is an upwinding parameter in the sense that λ > 0 results in an upwind
biased flux and λ = 0 results in a central flux. Castonguay et al. [52] demonstrated
that this choice for f⋆adv ensures Θadv ≤ 0.
The LDG formulation provides suitable expressions for u⋆ and f⋆dif (equations (3.97)
and (3.98)). Upon expanding equation (3.97), one obtains
u⋆ = uD − β · [[uD]]
=1
2
(uD− + uD+
)−(uD− − uD+
)(βxnx + βyny) , (3.115)
and upon substituting fdif = −bq into equation (3.98) and taking the dot product
with n, one obtains
f⋆dif · n =(fDdif+ τ [[uD]] + β[[fDdif ]]
)· n
=− b[1
2
(qDx− + qDx+
)nx +
1
2
(qDy− + qDy+
)ny
+((qDx− − qDx+
)nx +
(qDy− − qDy+
)ny
)(βxnx + βyny)
]+ τ
(uD− − uD+
),
(3.116)
where (on each face Γf)
[qDx+qDy+
]= qD
+ =
Nfp∑
j=1
qD(f,j)+ ℓ
1Df,j ,
[qDx−qDy−
]= qD
− =
Nfp∑
j=1
qD(f,j)− ℓ
1Df,j . (3.117)
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 64
In what follows, it will be shown that this choice for u⋆ and f⋆dif ensures Θdif ≤ 0.
On substituting equations (3.115) and (3.116) into the combination of equations (3.111)
and (3.112), and replacing uDk and qDk with uD− and qD
− = (qDx−, qDy−), respectively, one
obtains
1
2
d
dt‖UD‖2p,c
= −b ‖QD‖2p,κ +Θadv +
N∑
k=1
∫
Γk
[b
(uD−2
(qDx+nx + qDy+ny
)+uD+2
(qDx−nx + qDy−ny
)
+(uD+(qDx−nx + qDy−ny
)− uD−
(qDx+nx + qDy+ny
))(βxnx + βyny)
)+ τ
(uD−u
D+ − (uD−)
2)]dΓk
.
(3.118)
The last term on the RHS of equation (3.118) can be rewritten as a sum over edges
instead of a sum over elements. Assuming the domain is such that periodic bound-
ary conditions can be imposed, each edge receives contributions from two adjacent
elements. Using the notation uDe,+, qDxe,+
, qDye,+ and uDe,−, qDxe,−
, qDye,− to define the solu-
tion and auxiliary variable from the elements on the right and left sides of an edge,
respectively, equation (3.118) becomes
1
2
d
dt‖UD‖2p,c
= −b ‖QD‖2p,κ +Θadv +
Ne∑
e=1
∫
Γe
[b
(uDe,−2
(qDxe,+
nx + qDye,+ ny
)+uDe,+2
(qDxe,−
nx + qDye,− ny
)
− uDe,+2
(qDxe,−
nx + qDye,− ny
)− uDe,−
2
(qDxe,+
nx + qDye,+ ny
)
+(uDe,−
(qDxe,+
nx + qDye,+ ny
)− uDe,+
(qDxe,−
nx + qDye,− ny
))(βxnx + βyny)
+(uDe,+
(qDxe,−
nx + qDye,− ny
)− uDe,−
(qDxe,+
nx + qDye,+ ny
))(βxnx + βyny)
)
+ τ(uDe,−u
De,+ − (uDe,−)
2)+ τ
(uDe,+u
De,− − (uDe,+)
2)]dΓe
= −b ‖QD‖2p,κ +Θadv − τNe∑
e=1
∫
Γe
(uDe,+ − uDe,−
)2dΓe
. (3.119)
In the equations above, Γe represents edge e. Equation (3.99) follows immediately
from equation (3.119) because b ≥ 0, τ ≥ 0, and Θadv ≤ 0 from [52]. This completes
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 65
the proof of Theorem 3.6.1
Remark. Theorem 3.6.1 guarantees the stability of VCJH schemes for linear advection-
diffusion problems on triangles. In summary, this theorem was obtained assuming
that:
1. The approximate solution UD is computed on a domain of straight-sided trian-
gles, each with a constant value of Jk.
2. The correction functions (gf,j and hf,j) and fields (ψf,j and φf,j) are the VCJH
correction functions and fields defined in sections 3.3 and 3.4. This ensures that
equations (3.90) and (3.95) are satisfied, constants c and κ are non-negative,
and thus ‖UD‖p,c and ‖QD‖p,κ are norms.
3. The advective numerical flux is computed using the Lax-Friedrichs approach
(equation (3.113)).
4. The common solution and diffusive numerical flux are computed using the LDG
approach (equations (3.115) and (3.116)).
Note that, the stability proof has been constructed for the specific case in which the
Lax-Friedrichs approach is used for the advective numerical flux and the LDG ap-
proach is used for the common solution and diffusive numerical flux. However, the
proof still has broad applicability because the Lax-Friedrichs approach recovers the
central and upwind approaches, and the LDG approach recovers the BR1 and CF
approaches. Furthermore, it appears that the proof can be extended to alternative
flux formulations, in particular to compact approaches such as the CDG, BR2, or IP
approaches. However, such extensions are beyond the scope of this work because, as
discussed previously, this work is concerned with demonstrating the favorable per-
formance of VCJH discretizations paired with explicit time-stepping approaches, for
which compactness is not necessarily essential.
3.7 Linear Numerical Experiments
This section will investigate the orders of accuracy and explicit time-step limits asso-
ciated with the VCJH schemes on triangles by performing numerical experiments to
solve the linear advection-diffusion equation.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 66
Consider the 2D, time-dependent, linear advection-diffusion of a scalar u = u(x, t)
governed by equations (3.58) and (3.59) in the square domain [−1, 1]×[−1, 1], subjectto a sinusoidal initial condition u(x, 0) = sin(πx) sin(πy), and periodic boundary
conditions. The exact solution takes the form
ue = exp(−2bπ2t
)sin (π (x− axt)) sin (π (y − ayt)) , (3.120)
where ax = a cos θ and ay = a sin θ are the wave speeds in the x and y directions, and
b is the diffusivity coefficient. Numerical experiments were performed on two variants
of this problem: a diffusion problem with a = 0 and b = 0.1, and an advection-
diffusion problem with a = 1, b = 0.1, and θ = π/6. These problems were solved
using the VCJH schemes in conjunction with a Lax-Friedrichs formulation for the
advective numerical flux and a LDG formulation for the common solution and diffusive
numerical flux. The schemes were marched forward in time from t = 0 to t = 1 using
an explicit, low-storage, 5 stage, 4th order Runge-Kutta scheme for time advancement
(denoted RK54) [16]. For the order of accuracy analysis, the time-step was chosen
sufficiently small to ensure that temporal errors were negligible relative to spatial
errors.
In the experiments, VCJH schemes parameterized by c and κ were paired with Lax-
Friedrichs fluxes parameterized by λ and LDG fluxes parameterized by β and τ .
Values of c, κ, λ, β, and τ were selected as follows:
• Choosing c and κ: Four values of c, (namely, cdg, csd, chu, and c+), have been
shown to produce favorable results for advection problems on triangles [63]. For
advection problems on triangles, Castonguay et al. [63, 52] demonstrated that
cdg recovers a collocation-based nodal DG scheme (as mentioned previously),
csd and chu recover schemes with properties similar to the one-dimensional SD
scheme [48] and Huynh’s g2 scheme [23], and finally c+ recovers a scheme which
yields a maximum explicit time-step limit (as mentioned previously). In the
experiments documented in this work, these four values of c were paired with
Table 3.1: Reference values of c and κ for p = 2 and p = 3, for the 3-stage, 3rd-orderRunge-Kutta scheme (RK33), the 4-stage, 4th-order, Runge-Kutta scheme (RK44), and a5-stage, 4th-order, Runge-Kutta scheme (RK54) [16]. These values were computed usingvon Neumann analysis (similar to the analysis performed in [52]) of the VCJH schemes onuniform grids of right triangular elements.
Note that the experiments did not utilize values of c and κ that were larger
than c+ and κ+, as it has been shown that choosing c≫ c+ and κ≫ κ+ results
in a significant reduction in the order of accuracy of VCJH schemes for 1D
advection-diffusion problems [51].
• Choosing λ: All of the numerical experiments were performed with λ = 1.
This value of λ ensures that the advective numerical flux is computed using
information exclusively from the upwind direction.
• Choosing β: All of the numerical experiments were performed with β = ±0.5n−.
This value of β promotes compactness of the scheme (as mentioned previ-
ously) [53, 56].
• Choosing τ : All numerical experiments were performed with τ = 0.1. This
value of τ has been shown to yield favorable results for linear advection-diffusion
problems in 1D [51].
3.7.1 Orders of Accuracy and Explicit Time Step Limits
The orders of accuracy were evaluated on regular triangular meshes created by di-
viding cartesian N × N quadrilateral meshes into meshes with N = 2N2 triangular
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 68
elements. Triangular meshes with N = 16, 24, 32, 48, 64, 96, 128, and 192 were cre-
ated. On each mesh, errors in the solution and the solution gradient were measured
using the following L2 norm and seminorm
E(L2) =
√√√√N∑
k=1
∫
Ωk
(ue − uDk )2 dΩk, (3.121)
E(L2s) =
√√√√N∑
k=1
∫
Ωk
[(∂ue
∂x− ∂uDk
∂x
)2
+
(∂ue
∂y− ∂uDk
∂y
)2]dΩk. (3.122)
In equations (3.121) and (3.122), the integrals over each element domainΩk were com-
puted using a quadrature rule of sufficient strength. The expected order of accuracy
for E(L2) was p+ 1, and the expected order for E(L2s) was p.
In addition to orders of accuracy, explicit time-step limits were obtained for the VCJH
schemes. The explicit time-step limit for each scheme was determined by using an
iterative method to find the largest time-step that allowed the solution to remain
bounded at t = 1.
For each of the schemes, Tables (3.2) to (3.4) contain values of the time-step limit
and the absolute error obtained on the grid with N = 32, along with the order of
accuracy obtained on the sequence of grids described above.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 69
Table 3.2: VCJH scheme accuracy properties and explicit time-step limits for the modellinear advection-diffusion problem on triangles with a = 0, b = 0.1, and p = 2. Values ofλ = 1, β = ±0.5n−, and τ = 0.1 were used in the experiments. The time-step limit (∆tmax)and absolute errors (L2 and L2s err.) were obtained on the grid with N = 32.
Table 3.3: VCJH scheme accuracy properties and explicit time-step limits for the modellinear advection-diffusion problem on triangles with a = 0, b = 0.1, and p = 3. Values ofλ = 1, β = ±0.5n−, and τ = 0.1 were used in the experiments. The time-step limit (∆tmax)and absolute errors (L2 and L2s err.) were obtained on the grid with N = 32.
CHAPTER 3. ESFR FOR ADV-DIFF PROBLEMS ON TRIANGLES 70
Table 3.4: VCJH scheme accuracy properties and explicit time-step limits for the modellinear advection-diffusion problem on triangles with a = 1, b = 0.1, and p = 3. Values ofλ = 1, β = ±0.5n−, and τ = 0.1 were used in the experiments. The time-step limit (∆tmax)and absolute errors (L2 and L2s err.) were obtained on the grid with N = 32.
In particular, Tables (3.2) and (3.3) show the results for the diffusion problem, where
the order of accuracy was obtained on grids with N = 24, 32, 48, 64, 96, 128, and 192
for p = 2 and N = 16, 24, 32, 48, 64, 96, and 128 for p = 3. Table (3.4) shows the
results for the advection-diffusion problem, where the order of accuracy was obtained
on grids with N = 16, 24, 32, 48, 64, 96, and 128 for p = 3. Note that the results
for the diffusion and the advection-diffusion problems were similar, and thus limited
results (only the results for p = 3) are shown for the advection-diffusion problem.
The data in Tables (3.2) to (3.4) demonstrates that each scheme obtains the expected
order of accuracy. In addition, the data demonstrates that schemes with larger values
of c and κ have larger maximum time-steps. The best time-step improvements are
shown in Table (3.2) for p = 2, c = c+, and κ = κ+, where the maximum time-step is
approximately 2.2 times larger than that of the collocation-based nodal DG scheme.
Chapter 4
Energy Stable Flux Reconstruction
for Advection-Diffusion Problems on
Tetrahedra
71
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 72
4.1 Preamble
For the first time, this chapter constructs energy stable FR schemes (VCJH schemes)
for advection-diffusion problems on tetrahedra, and proves the stability of these
schemes for linear advection-diffusion problems for all orders of accuracy. Note
that this chapter is adapted from the article “Energy Stable Flux Reconstruction
for Advection-Diffusion Problems on Tetrahedra” by D. M. Williams and A. Jame-
son, which has been submitted to the Journal of Scientific Computing. The main
results and over 95% of the text for this article were contributed by D. M. Williams.
Due to the size of this chapter, it is useful to briefly examine its format. Section 2
of this chapter describes the general FR approach for treating tetrahedral elements,
section 3 introduces the energy stable FR approach (i.e. the VCJH approach) for
tetrahedral elements and proves the stability of this approach for linear advection-
diffusion problems, section 4 describes how to construct the VCJH correction fields,
section 5 identifies values of the coefficients c and κ that preserve the spatial symmetry
of the schemes, section 6 presents plots of the symmetric VCJH correction fields,
section 7 proves that the resulting class of VCJH schemes is equivalent to a class of
filtered DG schemes, section 8 describes how to construct the filtering matrices for
the filtered DG schemes, section 9 identifies the VCJH schemes which have explicit
time-step limits that are maximal (for linear advection problems), and finally, in an
effort to further assess the capabilities of the VCJH schemes, section 10 presents the
results of numerical experiments on a canonical linear advection-diffusion problem.
4.2 Flux Reconstruction for Advection-Diffusion
Problems on Tetrahedral Elements
In what follows, the general FR methodology for advection-diffusion problems on
triangular elements, described by Castonguay et al. ([51] and [52]) and Williams et
al. ([64] and Chapter 3 of this thesis), is extended to tetrahedral elements.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 73
4.2.1 Preliminaries
Consider the 3D advection-diffusion equation
∂u
∂t+∇ · f = 0, (4.1)
where u is a scalar solution, t is time, ∇ is the gradient operator defined such that∇ ≡(∂∂x, ∂∂y, ∂∂z
), x = (x, y, z) = (x1, x2, x3) are spatial coordinates, and f = f (u,∇u) is
an advective-diffusive flux. In keeping with the standard approach (from [58]), one
may eliminate second derivative terms from Equation (4.1) by rewriting it as the
following first-order system
∂u
∂t+∇ · f = 0, (4.2)
q−∇u = 0. (4.3)
This system depends on u and a new variable q that is commonly referred to as the
‘auxiliary variable.’ One seeks a solution to this system within the 3D domain Ω with
boundary Γ. Towards this end, one may assume that the domain can be divided into
N conforming, non-overlapping, straight-sided tetrahedral elements Ωk as follows
Ω =
N⋃
k=1
Ωk, (4.4)
Ωi ∩Ωj = ∅ ∀i 6= j. (4.5)
Within the kth element Ωk, the solution u can be approximated by a function uDkwhich is continuously defined within Ωk and which vanishes outside of the element.
The function uDk is labeled with a superscript D to indicate that, in general, it is
discontinuous at the boundary between neighboring elements Ωk and Ωk+1. In ac-
cordance with the traditional, discontinuous Finite Element approach (as described
in [13]), uDk can be approximated using a polynomial of degree p which takes the
following form
uDk =
Np∑
i=1
(uDk )i ℓi (x) , (4.6)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 74
where (uDk )i is the value of the solution at solution point i within Ωk and ℓi (x) is
the multi-dimensional nodal basis function which assumes the value of 1 at solution
point i and the value of 0 at all other solution points. Figure (4.1) shows an example
of the Np = 10 solution points which can be used to define a degree p = 2 polynomial
approximation of the solution on the tetrahedron.
(−1,1,−1)
(1,−1,−1)
(−1,−1,−1)
(−1,−1,1)
Figure 4.1: Example of the Np = 10 solution point locations (denoted by spheres) in thereference element for the case of p = 2.
In general, note that Np = (p+1)(p+2)(p+3)6
solution points are required to define a
degree p polynomial approximation of the solution on the tetrahedron.
In a similar fashion, the auxiliary variable q and the flux f within Ωk can be approx-
imated by vector-valued functions qDk and fDk (respectively) whose components are
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 75
polynomials of degree p which are defined as follows
qDk =
(qDxk, qDyk , q
Dzk
)=(qD1k , q
D2k, qD3k
),
∀ m = 1, 2, 3, qDmk=
Np∑
i=1
(qDmk)i ℓi (x) , (4.7)
fDk =(fDxk, fD
yk, fD
zk
)=(fD1k, fD
2k, fD
3k
),
∀ m, fDmk
=
Np∑
i=1
(fDmk
)i ℓi (x) , (4.8)
where (qDmk)i is the value of the mth component of the auxiliary variable at solution
point i, and (fDmk
)i is the value of the mth component of the flux at solution point i.
Note that, in general, the flux (fDmk
)i is a nonlinear function of both the solution (uDk )i
and the auxiliary variable (qDk )
i.
Upon substituting uDk , qDk , and fDk in place of u, q, and f in equations (4.2) and (4.3),
one obtains
∂uDk∂t
+∇ · fDk = 0, (4.9)
qDk −∇uDk = 0. (4.10)
As currently constructed, equations (4.9) and (4.10) involve only information that
is local to the kth element, as they relate the solution uDk in Ωk to the flux fDk and
auxiliary variable qDk in Ωk. In order to construct a valid numerical scheme, it is
necessary to incorporate information from neighboring elements. Towards this end,
the FR approach replaces the discontinuous flux fDk in equation (4.9) with a ‘recon-
structed flux’ fk of degree p + 1 which is required to be continuous in the following
sense: the normal components of fk and fk+1 are required to be equivalent to one
another on the boundary between neighboring elements Ωk and Ωk+1. In addition,
the FR approach replaces the discontinuous solution uDk in equation (4.10) with a
reconstructed solution uk of degree p + 1 which is required to be continuous in the
sense that uk and uk+1 are required to be equivalent to one another on the boundary
between Ωk and Ωk+1. Upon replacing fDk and uDk with fk and uk in equations (4.9)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 76
and (4.10), one obtains
∂uDk∂t
+∇ · fk = 0, (4.11)
qDk −∇uk = 0. (4.12)
Evidently, equations (4.11) and (4.12) represent a valid numerical scheme in the sense
that they utilize information from both the kth element and its neighbors.
Prior to solving equations (4.11) and (4.12), it is convenient to first transform these
equations from the domain of the physical element Ωk to the domain of a ‘reference
element’ ΩS. Towards this end, suppose that the element Ωk has vertices v1.k, v2,k,
v3,k, and v4,k, and the reference tetrahedronΩS has vertices (−1,−1,−1), (1,−1,−1),(−1, 1,−1), and (−1,−1, 1). One may define a mapping Θk between the physical
coordinates x in Ωk and the reference coordinates x = (x, y, z) in ΩS, as follows
x = Θk (x) = −x+ y + z + 1
2v1,k +
x+ 1
2v2,k +
y + 1
2v3,k +
z + 1
2v4,k. (4.13)
Using the mapping in equation (4.13), one may construct the Jacobian matrix Jk =
∇Θk (where ∇ =(
∂∂x, ∂∂y, ∂∂z
)) and the Jacobian determinant Jk = det(Jk). Next,
one may use Jk and Jk in order to transform the physical quantities uk, uDk , fk, and
qDk defined on Ωk into reference quantities u, uD, f , and qD defined on ΩS as follows
u = Jk uk (Θk (x) , t) , uD = Jk uDk (Θk (x) , t) , (4.14)
f = Jk J−1k fk, qD = Jk J
Tk q
Dk = ∇u = Jk J
Tk∇uk. (4.15)
Note that, as a result of equations (4.14) and (4.15), the following equations also hold
∇ · fk = J−1k
(∇ · f
), (4.16)
∇uk · fk = J−2k
(∇u · f
), (4.17)
∫
Ωk
uk (∇ · fk) dΩk = J−1k
∫
ΩS
u(∇ · f
)dΩS, (4.18)
∫
Ωk
∇uk · fk dΩk = J−1k
∫
ΩS
∇u · f dΩS, (4.19)
∫
Γk
uk (fk · n) dΓk = J−1k
∫
ΓS
u(f · n
)dΓS. (4.20)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 77
Equations (4.14) - (4.20) will be used frequently in subsequent discussions. For now,
consider substituting equations (4.14) - (4.16) into equations (4.11) and (4.12), in
order to obtain the following
∂uD
∂t+ ∇ · f = 0, (4.21)
qD − ∇u = 0. (4.22)
Observe that equations (4.21) and (4.22) are defined exclusively on the reference
domain ΩS, as they have been obtained by transforming equations (4.11) and (4.12)
from the physical domain Ωk to the reference domain ΩS. In what follows, the FR
procedure for solving equations (4.21) and (4.22) will be discussed.
4.2.2 The FR Procedure
The FR procedure for solving equations (4.21) and (4.22) involves obtaining the
unknown quantities qD, the auxiliary variable in reference space, and ∂uD
∂t, the time
rate of change of the solution in reference space, from computations of (respectively)
∇u, the gradient of the reconstructed solution in reference space, and ∇ · f , the
divergence of the reconstructed flux in reference space.
4.2.2.1 Computing ∇u, the Gradient of the Reconstructed Solution in
Reference Space
In order to facilitate the computation of ∇u, one must first formulate a more precise
definition for u. Towards this end, the FR approach requires u to take the following
form on the element boundary ΓS
u
∣∣∣∣ΓS
=(uD + uC
) ∣∣∣∣ΓS
, (4.23)
where uC is a ‘solution correction’ that corrects uD. On the element boundary, the
sum of uD and uC is required to assume the value of u⋆, the value of the common
numerical solution in reference space. This is equivalent to requiring that the following
expression holds
u
∣∣∣∣ΓS
= u⋆∣∣∣∣ΓS
=(uD + uC
) ∣∣∣∣ΓS
. (4.24)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 78
Equation (4.24) ensures that u (and thus uk) is continuous at the boundary between
neighboring elements Ωk and Ωk+1. In practice, equation (4.24) is enforced pointwise
at the l = 1, . . . , Nfp ‘flux points’ on the f = 1, . . . , Nef faces of the element, as
follows
uf,l = u⋆f,l = uDf,l + uCf,l ∀f, l (4.25)
where (for example) uDf,l is the value of uD at flux point l on face f .
For reference, the f = 1, . . . , 4 faces of the tetrahedron are shown in Figure (4.2), and
the l = 1, . . . , 6 flux points for the case of f = 1, p = 2 are shown in Figure (4.3).
f = 2 (back)
f = 1 (front)
f = 4 (bottom)
f = 3 (back)
Figure 4.2: Example of the numbering convention for the faces on the reference element.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 79
l = 3
l = 5
l = 2
l = 6
l = 4
l = 1
Figure 4.3: Example of the numbering convention for the flux points on the reference elementfor the case of p = 2. The flux points (denoted by squares) are shown for the face f = 1.
Note that, in general Nfp = (p+ 1)(p+ 2)/2 and Nef = 4 for a tetrahedron.
Now, having formulated u in terms of uD and uC via equations (4.23)-(4.25), one may
formulate ∇u in a similar fashion. In particular, one may express ∇u as follows
∇u = ∇uD + ∇uC , (4.26)
where ∇uD can be computed by applying the gradient operator ∇ to uD, and ∇uCcan be computed by applying the ‘lifting operators’ (or ‘correction fields’) ψf,l to
values of uCf,l as follows
∇uC (x) =
Nfe∑
f=1
Nfp∑
l=1
uCf,lnf,l ψf,l (x)
=
Nfe∑
f=1
Nfp∑
l=1
[u⋆f,l − uDf,l
]nf,l ψf,l (x) . (4.27)
The lifting operators ψf,l(x) are designed to ‘lift’ (or transform) values of uC defined
on ΓS into values of ∇uC defined onΩS. In order to ensure that the operators perform
this task, they are defined such that ψf,l ≡ ∇ · gf,l, where each gf,l is a vector-valued
‘correction function’ associated with flux point l on face f . In addition, the normal
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 80
component of the correction function (gf,l · n) is required to assume the value of 1 at
flux point l on face f and to assume the value of 0 at all neighboring flux points as
follows
gf,l (xv,w) · nv,w =
1 if v = f and w = l
0 if v 6= f or w 6= l. (4.28)
Furthermore, gf,l (x) is required to belong to the Raviart-Thomas space of degree p
(defined in [61]) in order to ensure that the following holds
ψf,l ≡ ∇ · gf,l ∈ Pp (ΩS) , gf,l · n ∈ Rp (ΓS) , (4.29)
where Pp (ΩS) and Rp (ΓS) are spaces which contain the polynomials of degree ≤ p on
ΩS and ΓS, respectively. In [64] (and Chapter 3 of this thesis), Williams et al. showed
that equations (4.28) and (4.29) ensure that ψf,l serves as a lifting operator that
transforms uC defined on ΓS into ∇uC defined on ΩS.
Figures (4.4) and (4.5) show an example of a vector correction function gf,l and an
associated correction field (lifting operator) ψf,l, for the case of p = 2.
Figure 4.4: Example of a vector correction function gf,l associated with flux pointf = 1, l = 2 for the case of p = 2.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 81
Figure 4.5: Contours of a correction field ψf,l (where ψf,l ≡ ∇ · gf,l) associated with fluxpoint f = 1, l = 2 for the case of p = 2.
4.2.2.2 Computing ∇ · f , the Divergence of the Reconstructed Flux in
Reference Space
One must now define a precise form for f in order to facilitate the computation of
∇ · f . Towards this end, f is required to take the following form in reference space
f = fD + fC , (4.30)
where fD is the discontinuous flux (fDk ) in reference space, and fC is a ‘corrective flux’
which corrects fD. On each face of the element, the sum of the normal components
of fD and fC is required to equal the normal component of f⋆, the common numerical
flux in reference space. Equivalently, the following expression is required to hold
f · n∣∣∣∣ΓS
= f⋆ · n∣∣∣∣ΓS
=(fD + fC
)· n∣∣∣∣ΓS
. (4.31)
Equation (4.31) ensures that the normal component of f (and thus fk) is continuous at
the boundary between neighboring elements. In practice, equation (4.31) is enforced
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 82
pointwise at the flux points on the element faces as follows
ff,l · nf,l = f⋆f,l · nf,l =(fDf,l + fCf,l
)· nf,l ∀f, l. (4.32)
Now, having obtained a definition of f in terms of fD and fC , one can obtain a similar
definition of ∇ · f . Towards this end, one may express ∇ · f as follows
∇ · f = ∇ · fD + ∇ · fC , (4.33)
where ∇ · fD is obtained by applying the divergence operator ∇· to fD, and ∇ · fCis obtained by applying lifting operators (correction fields) φf,l to values of fCf,l, as
follows
∇ · fC =
Nfe∑
f=1
Nfp∑
l=1
[fCf,l · nf,l
]φf,l
=
Nfe∑
f=1
Nfp∑
l=1
[(f⋆f,l − fDf,l
)· nf,l
]φf,l. (4.34)
In equation (4.34), each correction field φf,l is defined as the divergence of a vector
correction function hf,l. The functions hf,l are required to possess the same properties
as the functions gf,l (which were defined previously). This ensures that the resulting
FR schemes are conservative as shown by Castonguay et al. [52].
4.2.2.3 Obtaining a Final System of Equations
Upon substituting the expressions for ∇ · f (equations (4.33) and (4.34)) and ∇u(equations (4.26) and (4.27)) into equations (4.21) and (4.22), respectively, one ob-
tains a complete description of the FR approach on the reference element ΩS, as
follows
∂uD
∂t+ ∇ · fD + ∇ · fC
=∂uD
∂t+ ∇ · fD +
Nfe∑
f=1
Nfp∑
l=1
[(f⋆f,l − fDf,l
)· nf,l
]φf,l = 0, (4.35)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 83
and
qD − ∇uD − ∇uC
= qD − ∇uD −Nfe∑
f=1
Nfp∑
l=1
[u⋆f,l − uDf,l
]nf,l ψf,l = 0. (4.36)
Equations (4.35) and (4.36) can be evaluated at theNp solution points withinΩS in or-
der to yield 4Np equations for the 4Np unknowns (∂uD
∂t)1, . . . , (∂u
D
∂t)Np and (qD)1, . . . , (qD)Np.
The behavior of the FR scheme defined by equations (4.35) and (4.36) is determined
by six factors:
1. The locations of the solution points xi within the reference element.
2. The locations of the flux points xf,l on the boundary of the reference element.
3. The procedure for computing the common numerical solution values u⋆f,l in
reference space.
4. The procedure for computing the common numerical flux values f⋆f,l in reference
space.
5. The procedure for computing the solution correction fields ψf,l.
6. The procedure for computing the flux correction fields φf,l.
If appropriate procedures are chosen for computing the common numerical solution
values u⋆f,l in reference space, the common numerical flux values f⋆f,l in reference
space, the solution correction fields ψf,l, and the flux correction fields φf,l, the FR
schemes can be proven stable for linear advection-diffusion problems, independent of
the locations of the solution points xi and flux points xf,l. This will be shown in the
following section.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 84
4.3 Proof of Stability of VCJH Schemes for Linear
Advection-Diffusion Problems
In this section, it will be shown that if the solution and flux correction fields ψf,l
and φf,l are chosen to be the ‘VCJH correction fields’, and if the common numerical
solution values u⋆f,l and common numerical flux values f⋆f,l are chosen appropriately,
the ensuing FR schemes (the VCJH schemes) are stable for linear advection-diffusion
problems.
4.3.1 Preliminaries
In order to examine the stability of the FR approach, it is useful to reformulate the
approach on the physical element Ωk. On Ωk, equations (4.35) and (4.36) of the FR
approach can be expressed succinctly as follows
∂uDk∂t
+∇ · fDk +∇ · fCk = 0, (4.37)
qDk −∇uDk −∇uCk = 0, (4.38)
where the reference quantities (defined on ΩS) in equations (4.35) and (4.36) were
converted into physical quantities (defined on Ωk) in equations (4.37) and (4.38) via
the transformations in equations (4.14) - (4.16).
The FR approach for solving equations (4.37) and (4.38) is considered to be ‘energy
stable’ ifN∑
k=1
(d
dt‖uDk ‖2
)≤ 0, (4.39)
or equivalentlyN∑
k=1
(d
dt‖Uk‖2
M
)≤ 0, (4.40)
where Uk =[(uDk )
1 . . . (uDk )Np]T
is a vector containing the solution values,
‖Uk‖M = UTk Mk Uk (4.41)
is a matrix-based norm, and Mk is a symmetric positive-definite matrix. Note that
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 85
the precise definition of Mk will be given later on in this section.
In equation (4.40), the squared norm of the solution ‖Uk‖2M
characterizes the en-
ergy of the solution. Therefore, equation (4.40) is a condition that ensures ‘energy
stability’, because it insists that the time rate of change of the solution energy is
non-positive.
It will be shown that equation (4.40) holds for a particular class of FR schemes,
referred to as the VCJH schemes. The proof of stability of the VCJH schemes will
consist of lemmas and a final theorem proving the stability of the schemes. In par-
ticular, the lemmas will summarize intermediate results which will be obtained from
manipulating equations (4.37) and (4.38), and the theorem will combine these results
in order to prove that equation (4.40) holds.
Lemma 4.3.1. Given that equation (4.37) holds for all FR schemes and provided
that the flux correction functions and fields (hf,l and φf,l) are chosen to be the VCJH
correction functions and fields, the following result holds
∫
Ωk
∂uDk∂t
ℓj dΩk +1
VS
p+1∑
v=1
v∑
w=1
cvw
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk
+
∫
Ωk
(∇ · fDk
)ℓj dΩk = −
∫
Γk
(fCk · n
)ℓj dΓk, (4.42)
where VS is the volume of the reference element ΩS, each cvw is a constant which
parameterizes φf,l (and thus hf,l), and each D(p,v,w) is a derivative operator which
will be subsequently defined.
Proof. Consider defining a derivative operator D(p,v,w) of degree p as follows
D(p,v,w) (·) = ∂p (·)∂x(p−v+1)∂y(v−w)∂z(w−1)
, (4.43)
where v = 1, . . . , p and w = 1, . . . , v. Note that, upon substituting all possible
combinations of v and w into equation (4.43), one recovers the (p + 1)(p + 2)/2
distinct derivatives of degree p in 3D.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 86
One may apply D(p,v,w) to equation (4.37) as follows
∂
∂t
(D(p,v,w)
(uDk))
+ D(p,v,w)(∇ · fDk
)+ D(p,v,w)
(∇ · fCk
)
=∂
∂t
(D(p,v,w)
(uDk))
+ D(p,v,w)(∇ · fCk
)= 0, (4.44)
where terms involving derivatives of the physical quantities w.r.t the reference coor-
dinates (i.e. terms such as D(p,v,w)(uDk)) are computed via the chain rule, and where
D(p,v,w)(∇ · fDk
)= 0 because ∇ · fDk is a degree p− 1 polynomial.
On multiplying equation (4.44) by D(p,v,w) (ℓj) and integrating over Ωk, one obtains
the following
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk +
∫
Ωk
D(p,v,w)(∇ · fCk
)D(p,v,w) (ℓj) dΩk = 0.
(4.45)
Upon substituting equation (4.16) (with fC in place of f) into equation (4.45) and
defining ℓj ≡ Jk ℓj, one obtains
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk+
1
Jk
∫
Ωk
D(p,v,w)(∇ · fC
)D(p,v,w)
(ℓj
)dΩS = 0.
(4.46)
On noting that D(p,v,w)(∇ · fC
)and D(p,v,w)
(ℓj
)are constants because ∇ · fC and
ℓj are degree p polynomials, one obtains
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk +
VSJkD(p,v,w)
(∇ · fC
)D(p,v,w)
(ℓj
)= 0.
(4.47)
Consider multiplying equation (4.47) by constant coefficients cvw, and thereafter sum-
ming over v and w as follows
p+1∑
v=1
v∑
w=1
cvw
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk
+VSJk
p+1∑
v=1
v∑
w=1
cvwD(p,v,w)
(∇ · fC
)D(p,v,w)
(ℓj
)= 0. (4.48)
The double summation in equation (4.48) contains (p+1)(p+2)/2 terms, one for each
of the distinct (p+1)(p+2)/2 derivative operators of degree p that can be formed in
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 87
3D. As a result, the term of the form∑p+1
v=1
∑vw=1 cvwD
(p,v,w) (·) is a general, weighted
sum of all distinct derivatives of degree p.
Next, it should be noted that if fC and ∇ · fC are constructed using the VCJH
correction functions hf,l (which were discussed in section 4.2.2.1) and the VCJH
fields φf,l (which have yet to be precisely defined), the following identity holds
p+1∑
v=1
v∑
w=1
cvw D(p,v,w)
(∇ · fC
)D(p,v,w)
(ℓj
)=
∫
ΓS
(fC · n
)ℓj dΓS−
∫
ΩS
(∇ · fC
)ℓj dΩS.
(4.49)
Section 4.4 will discuss a procedure for constructing the fields φf,l so as to ensure that
equation (4.49) holds.
Upon substituting equation (4.49) into equation (4.48) and rearranging the result,
one obtains
1
VS
p+1∑
v=1
v∑
w=1
cvw
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk
+1
Jk
[∫
ΓS
(fC · n
)ℓj dΓS −
∫
ΩS
(∇ · fC
)ℓj dΩS
]= 0. (4.50)
On substituting equations (4.18) and (4.20) with fC , fCk , ℓj, and ℓj in place of f , fk,
u, and uk into equation (4.50), one obtains
1
VS
p+1∑
v=1
v∑
w=1
cvw
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk
+
∫
Γk
(fCk · n
)ℓj dΓk −
∫
Ωk
(∇ · fCk
)ℓj dΩk = 0. (4.51)
Setting equation (4.51) aside for the moment, consider multiplying equation (4.37)
by the test function ℓj and integrating over Ωk, as follows
∫
Ωk
∂uDk∂t
ℓj dΩk +
∫
Ωk
(∇ · fDk
)ℓj dΩk +
∫
Ωk
(∇ · fCk
)ℓj dΩk = 0. (4.52)
Upon summing equations (4.52) and (4.51), and rearranging the result, one obtains
equation (4.42). This completes the proof of Lemma 4.3.1.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 88
Lemma 4.3.2. Given that equation (4.38) holds for all FR schemes and provided
that the solution correction functions and fields (gf,l and ψf,l) are chosen to be the
VCJH correction functions and fields, the following result holds
∫
Ωk
qDk · Lm,j dΩk +
1
VS
p+1∑
v=1
v∑
w=1
κvw
∫
Ωk
D(p,v,w)(qDk
)· D(p,v,w) (Lm,j) dΩk
−∫
Ωk
∇uDk · Lm,j dΩk =
∫
Γk
uCk (Lm,j · n) dΓk, (4.53)
where each κvw is a constant which parameterizes ψf,l (and thus gf,l), and each Lm,j
is a vectorial generalization of the nodal basis function ℓj, i.e. L1,j = Lx,j ≡ (ℓj, 0, 0),
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 93
Now, equations (4.74) and (4.77) can be recast in matrix form as follows
(Mk +
JkNp
p+1∑
v=1
v∑
w=1
cvw
(D(p,v,w)
)TD(p,v,w)
)d
dtUk +
3∑
m=1
(amS
kmUk − bSk
mQmk
)
=
∫
Γk
[((auDk − bqD
k
)− (au− bq)⋆
)· n]L dΓk, (4.80)
∀m,(Mk +
JkNp
p+1∑
v=1
v∑
w=1
κvw
(D(p,v,w)
)TD(p,v,w)
)Qmk
− SkmUk
= −∫
Γk
(uDk − u⋆
)nm L dΓk, (4.81)
where Mk ∈ RNp×Np is the local mass matrix with entries
[Mk]ij=
∫
Ωk
ℓi (x) ℓj (x) dΩk =
∫
Ωk
ℓi ℓj dΩk, (4.82)
Skm ∈ R
Np×Np is the local stiffness matrix with entries
[Skm
]ij=
∫
Ωk
ℓi (x)∂ℓj (x)
∂xmdΩk =
∫
Ωk
ℓi∂ℓj∂xm
dΩk, (4.83)
D(p,v,w) is the matrix form of the derivative operator D(p,v,w) defined such that
[D(p,v,w)Uk
]i≡ D(p,v,w)
(uDk (x)
) ∣∣∣∣xi
=
[Np∑
=1
(uDk)D(p,v,w) (ℓ (x))
]
xi
, (4.84)
Uk =[(uDk )
1 . . . (uDk )Np]T
is a vector containing the solution values (as mentioned
previously), Qmk=[(qDmk
)1 . . . (qDmk)Np]T
are vectors containing the auxiliary variable
values, and L = L (x) =[ℓ1 (x) . . . ℓNp (x)
]Tis a vector containing the nodal basis
functions.
Equations (4.80) and (4.81) can be simplified by introducing the following block
matrix definitions (in terms of cartesian coordinates x, y, and z)
Mk =
Mk 0 00 Mk 00 0 Mk
, (4.85)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 94
Sk =[Skx Sk
y Skz
], Sk =
Skx
Sky
Skz
, (4.86)
Qk =
Qxk
Qyk
Qzk
, A =
axIayIazI
, N =
nxInyInzI
, (4.87)
where I ∈ RNp×Np. In addition, one may define the following matrices in order to
further simply the notation
Kk =JkNp
p+1∑
v=1
v∑
w=1
cvw
(D(p,v,w)
)TD(p,v,w), (4.88)
Kk =JkNp
p+1∑
v=1
v∑
w=1
κvw
(D(p,v,w)
)TD(p,v,w) 0 0
0(D(p,v,w)
)TD(p,v,w) 0
0 0(D(p,v,w)
)TD(p,v,w)
.
(4.89)
Upon substituting equations (4.85) - (4.89) into equations (4.80) and (4.81), one
obtains
(Mk +Kk
) d
dtUk + SkAUk − bSkQk
=
∫
Γk
[((auDk − bqD
k
)− (au− bq)⋆
)· n]L dΓk, (4.90)
(Mk +Kk
)Qk − SkUk = −
∫
Γk
(uDk − u⋆
)NL dΓk. (4.91)
One may now define ‘modified mass matrices’ as follows
Mk ≡(Mk +Kk
), (4.92)
Mk ≡(Mk +Kk
), (4.93)
where Mk and Mk are guaranteed to be symmetric positive-definite provided that Kk
and Kk are symmetric positive-semidefinite. Note that if one requires that cvw ≥ 0
and κvw ≥ 0, then Kk and Kk are (in fact) symmetric positive-semidefinite.
On substituting equations (4.92) and (4.93) into equations (4.90) and (4.91), one
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 95
obtains equations (4.70) and (4.71). This completes the proof of Lemma 4.3.3.
Theorem 4.3.1. If the VCJH schemes (for which Lemmas 4.3.1 – 4.3.3 hold) are
employed in conjunction with the Lax-Friedrichs formulation [65, 66] for the advective
numerical flux f⋆adv
f⋆adv = (au)⋆ = auD+ λ
2|a · n| [[uD]], (4.94)
for which 0 ≤ λ ≤ 1 is an upwinding parameter, · is an averaging operator, [[·]]is a differencing (jump) operator, and the LDG formulation [53] for the common
numerical solution u⋆ and diffusive numerical flux f⋆dif
u⋆ = uD − β · [[uD]], (4.95)
f⋆dif = (bq)⋆ = bqD+ τ [[uD]] + β b[[qD]], (4.96)
for which β = (βx, βy, βz) = (β1, β2, β3) is a directional parameter and τ ≥ 0 is a
penalty parameter, then it can be shown that the following result holds
N∑
k=1
(d
dt‖Uk‖2
M
)≤ 0. (4.97)
Proof. Consider multiplying equation (4.70) on the left by UTk and equation (4.71)
on the left by QTk in order to obtain
UTk Mk d
dtUk +UT
k SkAUk − bUTk SkQk
=
∫
Γk
[((auDk − bqD
k
)− (au− bq)⋆
)· n]uDk dΓk, (4.98)
QTk MkQk −QT
k SkUk = −∫
Γk
(uDk − u⋆
) (qDk · n
)dΓk, (4.99)
where the fact that uDk = UTk L and qD
k ·n = QTk NL has been used. Equations (4.98)
and (4.99) can be simplified by introducing the following identities
UTk Mk d
dtUk =
1
2
d
dt
(UT
k MkUk
)=
1
2
d
dt‖Uk‖2
M, (4.100)
QTk MkQk = ‖Qk‖2M, (4.101)
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 96
where, because Mk and Mk are symmetric positive-definite matrices, ‖Uk‖M and
‖Qk‖M are norms. Upon using equations (4.100) and (4.101) to simplify equa-
tions (4.98) and (4.99), one obtains
1
2
d
dt‖Uk‖2
M+UT
k SkAUk − bUTk SkQk
=
∫
Γk
[((auDk − bqD
k
)− (au− bq)⋆
)· n]uDk dΓk, (4.102)
‖Qk‖2M −QTk SkUk = −
∫
Γk
(uDk − u⋆
) (qDk · n
)dΓk. (4.103)
One may now multiply equation (4.103) by b and add the result to equation (4.102)
in order to obtain
1
2
d
dt‖Uk‖2
M+ b ‖Qk‖2M +UT
k SkAUk − b(UT
k SkQk +QTk SkUk
)
=
∫
Γk
[((auDk − bqD
k
)− (au− bq)⋆
)· n]uDk dΓk − b
∫
Γk
(uDk − u⋆
) (qDk · n
)dΓk.
(4.104)
Equation (4.104) can be simplified by noting that
UTk SkAUk
=
∫
Ωk
uDk ∇ ·(auDk
)dΩk =
1
2
∫
Ωk
∇ ·(a(uDk)2)
dΩk =1
2
∫
Γk
(uDk)2
(a · n) dΓk,
(4.105)
and that
UTk SkQk +QT
k SkUk
=
∫
Ωk
(uDk(∇ · qD
k
)+ qD
k · ∇uDk)dΩk =
∫
Γk
uDk(qDk · n
)dΓk. (4.106)
Upon substituting equations (4.105) and (4.106) into equation (4.104) and rearranging
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 97
the result, one obtains
1
2
d
dt‖Uk‖2
M+ b ‖Qk‖2M
=
∫
Γk
[uDk
((auDk2− bqD
k
)− (au− bq)⋆
)+ b u⋆qD
k
]· n dΓk
=
∫
Γk
[uDk
(auDk2− (au)⋆
)− uDk
(bqD
k − (bq)⋆)+ b u⋆qD
k
]· n dΓk, (4.107)
where the fact that (au− bq)⋆ = (au)⋆− (bq)⋆ has been used. Next, upon summing
equation (4.107) over all elements in the mesh, one obtains
1
2
N∑
k=1
(d
dt‖Uk‖2
M
)
=− bN∑
k=1
‖Qk‖2M +
N∑
k=1
∫
Γk
[uDk
(auDk2− (au)⋆
)− uDk
(bqD
k − (bq)⋆)+ b u⋆qD
k
]· n dΓk
.
(4.108)
In order to demonstrate that the VCJH schemes are stable (in accordance with equa-
tion (4.40)), one must show that the right hand side (RHS) of equation (4.108) is
non-positive. The first term on the RHS of equation (4.108) is clearly non-positive,
however, the second term on the RHS has an ambiguous sign, and is only assured to
be non-positive for appropriate formulations of the numerical fluxes (au)⋆ and (bq)⋆,
and the common numerical solution u⋆. The Lax-Friedrichs approach [65, 66] is a
well-known approach for treating the advective numerical flux (au)⋆ and the Cen-
Table 4.1: Values of c+ for the 3-stage, 3rd-order RK scheme (RK33), the 4-stage, 4th-orderRK scheme (RK44), and a 5-stage, 4th-order RK scheme (RK54) [16].
The best results are obtained for the RK54 time-stepping scheme, for p = 2 and
c = c+ = 3.07×10−2. In this case, the time-step limit is 1.81x larger than that of the
collocation-based nodal DG scheme (for which c = 0).
4.10 Numerical Experiments on the Linear Advection-
Diffusion Equation
In an effort to further evaluate the performance of the VCJH schemes, they were
employed to solve a model linear advection-diffusion problem, and their orders of
accuracy and explicit time-step limits were computed.
The governing equation for the model linear advection-diffusion problem took the
following form∂u
∂t+∇ · (au) = b∆u, (4.206)
where the wave velocity vector a and the diffusion coefficient b were given the following
values: a = (ax, ay, az) =(1/2, 1/2,
√2/2)and b = 0.01. Approximate solutions to
equation (4.206) were sought within the domain Ω = [−1, 1]× [−1, 1]× [−1, 1], withperiodic boundary conditions imposed. At time t = 0, the solution was initialized to
u = sin (πx) sin (πy) sin (πz) throughout Ω, and thereafter, the solution was marched
forward in time to t = 1 using the explicit RK54 time-stepping scheme. At each
time-step, the advective numerical fluxes were computed using the Lax-Friedrichs
approach with λ = 1 and the common solutions and diffusive numerical fluxes were
computed using the LDG approach with τ = 1 and β = ±0.5n−.
The numerical experiments (described above) were used to evaluate the accuracy and
stability properties of two ‘representative’ VCJH schemes: the collocation-based nodal
DG scheme with c = 0 and κ = 0, and the scheme with c = c+ and κ = κ+ = c+.
The orders of accuracy and explicit time-step limits of these two VCJH schemes were
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 127
computed as follows:
1. The orders of accuracy were computed for p = 2 to p = 5 on a sequence of
regular tetrahedral grids. These grids were obtained using the approach from
section 4.9, i.e. by dividing the domain Ω into N3 cubic elements of equal size,
and then dividing each of these elements into 6 tetrahedral elements of equal
size, resulting in grids with a total of N = 6N3 tetrahedral elements. Grids with
N = 4, 6, 8, 12, 16, 24, 32, 48, and 64 were created, and on these grids, the
accuracy of the approximate solution was evaluated at t = 1 using the L2 norm
of the error between the approximate solution and the exact solution, where the
exact solution took the following form
uexact = exp(−3 b π2t
)sin (π (x− axt)) sin (π (y − ayt)) sin (π (z − azt)) .
The approximate solution was expected to converge towards the exact solution
at a rate of hp+1, where h was the mesh spacing.
In a similar fashion, the accuracy of the solution gradient was evaluated at t = 1
using the L2 norm of the error between the approximate solution gradient and
the exact solution gradient. In this case, the approximate solution gradient was
expected to converge towards the exact solution gradient at a rate of hp.
Note that, in each of the numerical experiments used to determine the rates of
convergence of the solution and its gradient, the time-steps were chosen small
enough to avoid effecting the spatial accuracy of the schemes.
Finally, note that due to the high accuracy of the schemes with p = 4 and
p = 5, reasonable results were obtained by computing the error on the coarser
meshes with N = 4, 6, 8, 12, 16, and 24 while neglecting the finest meshes with
N = 32, 48, and 64.
2. The explicit time-step limits were computed for p = 2 and p = 3 on the grid
with N = 64, and for p = 4 and p = 5 on the grid with N = 24. In each case,
the maximum time-step limit was taken to be the maximum time-step for which
the solution remained bounded at t = 1000∆t (i.e. boundedness was evaluated
after the first 1000 time-steps). Using this criterion, each maximum time-step
limit was determined via an iterative process, which computed the maximum
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 128
time-step to within an estimated tolerance of 1%.
The orders of accuracy and explicit time-step limits of the VCJH schemes are pre-
sented in Tables (4.2) and (4.3).
cdg, κdg c+, κ+
p N L2 u error O(L2) L2 ∇u error O(L2) L2 u error O(L2) L2 ∇u error O(L2)
Table 4.2: Accuracy of VCJH schemes for the advection-diffusion problem with a =(1/2, 1/2,
√2/2)and b = 0.01 on the grids with N = 4 to N = 64. The advective numerical
flux was computed using a Lax-Friedrichs flux with λ = 1 and the diffusive numerical fluxwas computed using a LDG flux with τ = 1 and β = ±0.5n−.
CHAPTER 4. ESFR FOR ADV-DIFF PROBLEMS ON TETRAHEDRA 129
cdg, κdg c+, κ+
p N ∆t′lim ∆t′lim2 64 3.24e-04 6.66e-043 64 1.51e-04 2.45e-044 24 4.72e-04 6.94e-045 24 2.74e-04 3.76e-04
Table 4.3: Time-step limits of VCJH schemes for the advection-diffusion problem witha =
(1/2, 1/2,
√2/2)and b = 0.01 on the grids with N = 24 and 64. The advective
numerical flux was computed using a Lax-Friedrichs flux with λ = 1 and the diffusivenumerical flux was computed using a LDG flux with τ = 1 and β = ±0.5n−.
Table (4.2) demonstrates that the expected order of accuracy is obtained by both the
scheme with c = 0 and κ = 0 and the scheme with c = c+ and κ = κ+, for all values
of p, except for p = 5 for which roundoff errors appear to have effected the results.
In addition, note that for p = 2 and p = 3, the scheme with c = c+ and κ = κ+
appears to be superconvergent, in some cases achieving more than 1/2 an order of
accuracy above that which was expected. Table (4.3) demonstrates that the scheme
with c = c+ and κ = κ+ has the larger explicit time-step limits. In particular, for
p = 2, the scheme with c = c+ and κ = κ+ has an explicit time-step limit which is
2.06x larger than that of the collocation-based nodal DG scheme (with c = 0 and
κ = 0).
Chapter 5
Reformulation of Energy Stable Flux
Reconstruction on Triangles
130
CHAPTER 5. REFORMULATION OF ESFR ON TRIANGLES 131
5.1 Preamble
In this chapter, the energy stable FR schemes (VCJH schemes) on triangles, originally
presented in chapter 3, are reformulated to incorporate the advances in theory that
were presented in chapter 4. In particular, in the previous chapter it was shown that
a certain class of parameterizing coefficients cvw and κvw ensure symmetry of the
VCJH derivative operators on tetrahedra. Furthermore, it was shown that the VCJH
schemes on tetrahedra can be formulated as a class of stable, filtered DG schemes. In
what follows, these results will be extended to the VCJH schemes on triangles.
5.2 Choosing Parameterizing Coefficients on Tri-
angles
The VCJH schemes on triangles employ degree p derivative operators, and it is im-
portant (as discussed previously) to ensure that these operators are symmetric. For
triangles, one may ensure symmetry by enforcing constraints on the coefficients as-
sociated with the derivative operators. More precisely, one may ensure symmetry
by constraining the coefficients cv and κv that appear in the following derivative
operatorsp+1∑
v=1
cv D(p,v) (f) D(p,v) (g) , (5.1)
andp+1∑
v=1
κv D(p,v) (F) · D(p,v) (G) , (5.2)
where f and g are arbitrary, continuous, p times differentiable, 2D scalar-valued
functions, and F and G are arbitrary, continuous, p times differentiable, 2D vector-
valued functions. The operators in equations (5.1) and (5.2) can be rewritten in
quadratic form as followsp+1∑
v=1
cv
(D(p,v) (f)
)2, (5.3)
andp+1∑
v=1
κv D(p,v) (F) · D(p,v) (F) . (5.4)
CHAPTER 5. REFORMULATION OF ESFR ON TRIANGLES 132
Now, having obtained the quadratic formulation of the VCJH derivative operators,
one may follow the approach of section 4.5, and construct a general, symmetric,
degree p derivative operator with which to compare the VCJH derivative operators.
Recall, that a general, symmetric, derivative operator of degree p was defined in
equation (4.143). Evidently, for d = 2, this operator can be employed on triangles.
For example, when d = 2 and p = 2, ∆f in equation (4.143) can be defined as follows
∆f =
[∂2f
∂x2,∂2f
∂y2,∂2f
∂x∂y,∂2f
∂y∂x
]T. (5.5)
Upon expanding ∆fT (∆f) in equation (4.143), one obtains an expression which, for
p > 1, contains non-distinct derivative terms, e.g. for d = 2 and p = 2, one obtains
an expression with c(
∂2f∂x∂y
)2and c
(∂2f∂y∂x
)2. More generally, for d = 2 and p ≥ 1, the
expanded form of equation (4.143) contains a total of
(p
v − 1
)(5.6)
derivative terms of the form
c
(∂p (·)
∂x(p−v+1)∂y(v−1)
)2
. (5.7)
Here, it is interesting to note that equation (5.6) contains an expression for the pth
degree binomial coefficients.
Upon combining all non-distinct derivative terms, one obtains the following operator
with p+ 1 distinct derivative terms
c
p+1∑
v=1
(p
v − 1
)(∂p (·)
∂x(p−v+1)∂y(v−1)
)2
, (5.8)
or equivalently
c
p+1∑
v=1
(p
v − 1
)(D(p,v) (f)
)2. (5.9)
The symmetric derivative operator in equation (5.9) can now be compared to the
CHAPTER 5. REFORMULATION OF ESFR ON TRIANGLES 133
VCJH derivative operator in equation (5.3). From this comparison, it immediately
follows that in order for the VCJH operator to be symmetric, one must require the
coefficients cv to be defined as follows
cv = c
(p
v − 1
). (5.10)
In addition, upon substituting F = (F1,F2) into equation (5.4), substituting κ and
Fm (for m = 1, 2) into equation (5.9) (in place of c and f), and comparing the
resulting expressions, one finds that the coefficients κv must be defined as follows
κv = κ
(p
v − 1
). (5.11)
One may observe that the coefficients in equations (5.10) and (5.11) are the same that
were recommended for the VCJH schemes on triangles in sections 3.3 and 3.4. The
previous recommendation was based on the numerical experiments by Castonguay
et al. [52] which identified coefficients cv and κv that ensured the symmetry of the
VCJH operators for p = 1 to p = 6. In this section, the author has succeeded in
confirming the results of Castonguay et al. by presenting an analytical derivation of
the required coefficients. This derivation extends the original results, proving that
the desired form of the coefficients can be obtained for all p ≥ 1.
5.3 Proof of Stability and Formulation of a Class
of Filtered DG Schemes on Triangles
In section 3.6, a class of VCJH schemes on triangles was proven to be stable using
a vector-calculus-based approach; it turns out, that the same schemes can be proven
to be stable via the matrix-based approach that was utilized to prove the stability of
the VCJH schemes on tetrahedra in section 4.3. In fact, the matrix-based proof of
the stability of the VCJH schemes on triangles is virtually identical to the proof on
tetrahedra, with the exception that equations (4.49) and (4.65) now take the following
CHAPTER 5. REFORMULATION OF ESFR ON TRIANGLES 134
form
p+1∑
v=1
cv D(p,v)
(∇ · fC
)D(p,v)
(ℓj
)=
∫
ΓS
(fC · n
)ℓj dΓS −
∫
ΩS
(∇ · fC
)ℓj dΩS,
(5.12)
p+1∑
v=1
κv D(p,v)
(∇uC
)· D(p,v)
(Lm,j
)=
∫
ΓS
uC(Lm,j · n
)dΓS −
∫
ΩS
∇uC · Lm,j dΩS,
(5.13)
where ΩS denotes the domain of the reference right triangle.
In addition, it is useful to note that the VCJH schemes on triangles are equivalent to
a class of filtered, collocation-based, nodal DG schemes. For these schemes, the filter
matrices F1 and F2 are defined by equation (4.176), and the associated matrices Kk
and Kk are defined as follows
Kk =JkNp
p+1∑
v=1
cv
(D(p,v)
)TD(p,v), (5.14)
Kk =JkNp
p+1∑
v=1
κv
(D(p,v)
)TD(p,v) 0
0(D(p,v)
)TD(p,v)
, (5.15)
where
[D(p,v)]ij = D(p,v) (ℓj (x))
∣∣∣∣xi
. (5.16)
Part II
Stability Theory for Nonlinear
Advection-Diffusion Problems
135
Chapter 6
Energy Stable Flux Reconstruction
for Nonlinear Advection-Diffusion
Problems on Tetrahedra
136
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 137
6.1 Preamble
In this chapter, the stability of the ESFR (VCJH) schemes for nonlinear advection-
diffusion problems on tetrahedra is examined. It is shown that the stability of the
schemes is primarily effected by the following factors: the degree of nonlinearity of the
diffusive flux, the methods for computing the advective numerical flux, the common
solution, and the diffusive numerical flux, and the choices of locations for the solution
points and flux points. In particular, it is shown that the degree of nonlinearity
of the diffusion coefficient b has a critical effect on determining the boundedness
of contributions from the diffusive flux, that the methods for forming the common
solution and numerical fluxes have a critical effect on the signs of contributions from
the interfaces, and that the locations of the solution points and flux points have a
critical effect on so-called ‘aliasing’ errors that arise from the schemes’ attempts to
represent nonlinear advective-diffusive fluxes, which are generally infinite dimensional,
with finite dimensional polynomial bases.
6.2 Preliminaries
Consider the general 3D nonlinear advection-diffusion equation that takes the follow-
ing form∂u
∂t+∇ · (fadv (u) + fdif (u,∇u)) = 0. (6.1)
In order to facilitate the analysis, it is convenient to follow the approach of [53] and
to consider the following simplified version of equation (6.1)
∂u
∂t+∇ · (fadv (u)− b (u)∇u) = 0, (6.2)
where b (u) ≥ 0. Note that the non-negativity of b (u) is necessary to ensure that the
problem is well-posed.
In accordance with the standard approach, equation (6.2) can be rewritten as a first
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 138
order system in order to eliminate second derivative terms as follows
∂u
∂t+∇ · (fadv (u)− b (u)q) = 0, (6.3)
q−∇u = 0. (6.4)
In the following section, the stability of the VCJH schemes for solving equations (6.3)
and (6.4) will be examined.
6.3 Stability Analysis
In order to facilitate the analysis, the VCJH schemes for nonlinear advection-diffusion
can be cast in matrix form, as shown in the following lemma.
Lemma 6.3.1. Suppose that the VCJH schemes are employed to solve the nonlinear
advection-diffusion equation for which the flux f takes the following form
f = fadv + fdif = fadv (u)− b (u)q. (6.5)
Then the following formulation of the VCJH schemes (in terms of matrices) holds
Mk d
dtUk + Sk (Fk, adv + Fk, dif )
=
∫
Γk
[((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n]L dΓk, (6.6)
Mk Qk − SkUk = −∫
Γk
(uDk − u⋆
)NL dΓk, (6.7)
where Fk, adv and Fk, dif are matrices containing the x, y, and z components of the
advective and diffusive fluxes at the Np solution points, and Uk, Qk, Mk, Mk, Sk,
Sk, N, and L are matrices which were defined previously in Chapter 4.
Proof. Equation (6.7) was derived in Lemma 4.3.3, but the derivation of equation (6.6)
remains to be shown. In order to derive equation (6.6), consider substituting the
expressions for the discontinuous flux fDk = fDk, adv + fDk, dif and the flux correction
fCk = f⋆− fDk =(f⋆adv + f⋆dif
)−(fDk, adv + fDk, dif
)into equation (4.42), and insisting that
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 139
the result holds for all j in order to obtain
∀j∫
Ωk
∂uDk∂t
ℓj dΩk +1
VS
p+1∑
v=1
v∑
w=1
cvw
∫
Ωk
∂
∂t
(D(p,v,w)
(uDk))D(p,v,w) (ℓj) dΩk
+
∫
Ωk
(∇ ·(fDk, adv + fDk, dif
))ℓj dΩk =
∫
Γk
((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n ℓj dΓk.
(6.8)
Next, the second integral on the RHS of equation (6.8) can be transformed from an
integral over Ωk into an integral over ΩS as follows
∀j∫
Ωk
∂uDk∂t
ℓj dΩk +JkVS
p+1∑
v=1
v∑
w=1
cvw
∫
ΩS
∂
∂t
(D(p,v,w)
(uDk (x)
))D(p,v,w) (ℓj (x)) dΩS
+
∫
Ωk
(∇ ·(fDk, adv + fDk, dif
))ℓj dΩk =
∫
Γk
((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n ℓj dΓk,
(6.9)
where uDk (x) and ℓj (x) have been defined using the mapping Θk (x) (in accordance
with equations (4.75) and (4.76)). Equation (6.9) can now be rewritten in matrix
form as follows
Mk d
dtUk +
3∑
m=1
(SkmFmk , adv + Sk
mFmk , dif
)
=
∫
Γk
[((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n]L dΓk, (6.10)
where Fmk , adv and Fmk , dif are vectors which contain the values of the advective and
diffusive fluxes at the Np solution points, i.e.
Fmk , adv =[(fDmk , adv
)1. . .(fDmk , adv
)Np]T, (6.11)
Fmk , dif =[(fDmk , dif
)1. . .(fDmk , dif
)Np]T
= −[b((uDk)1) (
qDmk
)1. . . b
((uDk)Np) (qDmk
)Np]T. (6.12)
Equation (6.10) can be simplified by introducing the following block matrix definitions
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 140
which are written in terms of cartesian coordinates x, y, and z
Fk, adv =
Fxk, adv
Fyk , adv
Fzk , adv
, Fk, dif =
Fxk, dif
Fyk, dif
Fzk, dif
, (6.13)
where one should note that (for example) Fxk, adv = F1k , adv and Fxk, dif = F1k , dif .
Upon substituting equations (4.86) and (6.13) into equation (6.10), one obtains equa-
tion (6.6). This completes the proof of Lemma 6.3.1.
Theorem 6.3.1. Suppose that in the limit of a weakly nonlinear flux or infinite mesh
resolution, the diffusion coefficients become element-wise constant, i.e. b(uDk)≈ bk
(to within machine precision), and therefore a matrix of the diffusion coefficients Bkcan be defined as follows
Bk = bkI. (6.14)
Then it can be shown that
N∑
k=1
(QT
kBkMk Qk
)≥ 0. (6.15)
Subject to these conditions, if the VCJH schemes (for which Lemma 6.3.1 holds) are
employed in conjunction with an ‘E-flux’ formulation of the advective numerical flux
f⋆adv for which the following holds
(uDf+ − uDf−
)(f⋆adv − fadv (ξ)) · n ≤ 0, (6.16)
where ξ ∈[min
(uDf−, u
Df+
),max
(uDf−, u
Df+
)], a formulation of the common numerical
solution u⋆ and diffusive numerical flux f⋆dif for which the following holds
[ (uDf+ − uDf−
)f⋆dif + u⋆
(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))
+ uDf−fdif(uDf−,q
Df−
)− uDf+fdif
(uDf+,q
Df+
) ]· n ≤ 0, (6.17)
and exact L2 projections of the nonlinear fluxes fDk, adv and fDk, dif for which all aliasing
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 141
errors vanish, then it can be shown that
N∑
k=1
(d
dt‖Uk‖2
M
)≤ 0. (6.18)
Proof. Consider defining the matrices Bk and Bk as follows
Bk =
b((uDk)1)
0 · · · · · · 0
0. . .
......
. . ....
.... . . 0
0 · · · · · · 0 b((uDk)Np)
, (6.19)
Bk =
Bk 0 00 Bk 00 0 Bk
. (6.20)
One may now define Fk, dif as
Fk, dif = −BkQk. (6.21)
This definition of Fk, dif can be used to rewrite equation (6.7). In particular, one may
multiply equation (6.7) on the left by QTkBk and substitute equation (6.21) into the
result in order to obtain
QTkBkMk Qk + FT
k, dif SkUk =
∫
Γk
(uDk − u⋆
) (fDk, dif · n
)dΓk, (6.22)
where the fact that fDk, dif · n = FTk, dif NL has been used.
Setting equation (6.22) aside for the moment, consider multiplying equation (6.6) on
the left by UTk and substituting equation (4.100) into the result in order to obtain
1
2
d
dt‖Uk‖2
M+UT
k Sk (Fk, adv + Fk, dif )
=
∫
Γk
[((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n]uDk dΓk, (6.23)
where the fact that uDk = UTk L has been used. Upon summing equations (6.22) and
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 142
(6.23) and rearranging the result, one obtains
1
2
d
dt‖Uk‖2
M+QT
kBkMk Qk +UTk SkFk, adv +
(UT
k SkFk, dif + FTk, dif SkUk
)
=
∫
Γk
[((fDk, adv + fDk, dif
)−(f⋆adv + f⋆dif
))· n]uDk dΓk +
∫
Γk
(uDk − u⋆
) (fDk, dif · n
)dΓk.
(6.24)
Equation (6.24) can be simplified by noting that
UTk SkFk, dif + FT
k, dif SkUk
=
∫
Ωk
(uDk(∇ · fDk, dif
)+ fDk, dif · ∇uDk
)dΩk =
∫
Γk
uDk(fDk, dif · n
)dΓk. (6.25)
On substituting equation (6.25) into equation (6.24), one obtains
1
2
d
dt‖Uk‖2
M+QT
kBkMk Qk +UTk SkFk, adv
=
∫
Γk
[(fDk, adv −
(f⋆adv + f⋆dif
))· n]uDk dΓk +
∫
Γk
(uDk − u⋆
) (fDk, dif · n
)dΓk. (6.26)
Now, one may simplify equation (6.26) by manipulating the third term on the LHS.
One may expand this term as follows
UTk SkFk, adv =
∫
Ωk
(∇ · fDk, adv
)uDk dΩk
=
∫
Γk
(fDk, adv · n
)uDk dΓk −
∫
Ωk
∇uDk · fDk, adv dΩk. (6.27)
Upon substituting equation (6.27) into equation (6.26) and rearranging the result,
one obtains
1
2
d
dt‖Uk‖2
M+QT
kBkMk Qk
=
∫
Ωk
∇uDk · fDk, adv dΩk −∫
Γk
(f⋆adv · n) uDk dΓk +
∫
Γk
[(uDk − u⋆
)fDk, dif − uDk f⋆dif
]· n dΓk.
(6.28)
Next, one may consider adding and subtracting the following expression from the
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 143
RHS of equation (6.28)
∫
Ωk
∇uDk · fadv(uDk)dΩk +
∫
Γk
[(uDk − u⋆
)fdif(uDk ,q
Dk
)]· n dΓk, (6.29)
in order to obtain
1
2
d
dt‖Uk‖2
M+QT
kBkMk Qk
=
∫
Ωk
∇uDk · fadv(uDk)dΩk −
∫
Γk
(f⋆adv · n) uDk dΓk
+
∫
Γk
[(uDk − u⋆
)fdif(uDk ,q
Dk
)− uDk f⋆dif
]· n dΓk + εΩk
+ εΓk, (6.30)
where εΩkand εΓk
are defined as follows
εΩk≡∫
Ωk
∇uDk ·(fDk, adv − fadv
(uDk))dΩk, (6.31)
εΓk≡∫
Γk
[(uDk − u⋆
) (fDk, dif − fdif
(uDk ,q
Dk
))]· n dΓk. (6.32)
Observe that εΩkand εΓk
are measures of the effective ‘distances’ between the exact
flux functions (fadv(uDk)and fdif
(uDk ,q
Dk
)) and the approximate fluxes (fDk, adv and
fDk, dif ).
Upon summing equation (6.30) over all N elements in the mesh, one obtains
1
2
N∑
k=1
(d
dt‖Uk‖2
M
)+
N∑
k=1
(QT
kBkMk Qk
)= Ξadv + Ξdif + Ξalias, (6.33)
where Ξadv and Ξdif denote contributions from the advective and diffusive fluxes that
are defined as follows
Ξadv ≡N∑
k=1
∫
Ωk
∇uDk · fadv(uDk)dΩk −
∫
Γk
(f⋆adv · n) uDk dΓk
, (6.34)
Ξdif ≡N∑
k=1
∫
Γk
[(uDk − u⋆
)fdif(uDk ,q
Dk
)− uDk f⋆dif
]· n dΓk
, (6.35)
and where Ξalias denotes the contribution from the aliasing error that is defined as
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 144
follows
Ξalias ≡N∑
k=1
(εΩk+ εΓk
) . (6.36)
Stability is ensured if the time rate of change of the norm on the LHS of equation (6.33)
is non-positive. In other words, stability is ensured if Ξadv ≤ 0, Ξdif ≤ 0, Ξalias ≤ 0,
andN∑
k=1
(QT
kBkMk Qk
)≥ 0. (6.37)
Unfortunately, the condition in equation (6.37) is difficult to satisfy, as it requires
BkMk to be a symmetric positive-definite matrix. In all cases, BkMk is at least
‘weakly’ positive-definite (according to [74]), as it can be written as the product of
‘strongly’ positive-definite matrices, where ‘strong positive-definiteness’ is equivalent
to the conventional notion of positive-definiteness. However, for a general b (u), BkMk
need not be symmetric or positive-definite. In fact, BkMk is only guaranteed to be
symmetric positive-definite if Bk and Mk commute, and if Bk and Mk are simulta-
neously diagonalizable (i.e. Bk and Mk possess the same eigenvectors). This occurs
when
Bk = bkI, (6.38)
where bk is a constant and (from before) I is the identity matrix in R3Np×3Np . In
practice, equation (6.38) can be satisfied (to within machine precision) if b(uDk)≈ bk.
This happens for all nonlinear functions b (u) in the limit of infinite mesh resolution,
and for some weakly nonlinear functions b (u) for coarser mesh resolutions.
Now, having determined the conditions under which equation (6.37) holds, one must
establish the conditions under which Ξadv ≤ 0, Ξdif ≤ 0, and Ξalias ≤ 0, in order to
complete the proof.
It turns out that if the advective numerical flux f⋆adv is chosen to be an ‘E-flux’ then
Ξadv ≤ 0. This can be seen as follows. Consider defining a vector-valued function
G (u) such that∂G
∂u(u) =
∂G
∂u= fadv. (6.39)
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 145
One may now substitute equation (6.39) into equation (6.34) in order to obtain
Ξadv =
N∑
k=1
∫
Ωk
∇uDk ·∂G
∂u
(uDk)dΩk −
∫
Γk
(f⋆adv · n)uDk dΓk
=N∑
k=1
∫
Ωk
∇ ·G(uDk)dΩk −
∫
Γk
(f⋆adv · n) uDk dΓk
. (6.40)
Upon applying the divergence theorem to the first term on the RHS of equation (6.40)
and rearranging the result, one obtains
Ξadv =
N∑
k=1
∫
Γk
[G(uDk)− uDk f⋆adv
]· n dΓk
. (6.41)
Assuming that the domain Ω is such that periodic boundary conditions can be im-
posed, one may rewrite equation (6.41) in terms of a summation over element faces
f as follows
Ξadv =
Nf∑
f=1
∫
Γf
[G(uDf−)−G
(uDf+)+(uDf+ − uDf−
)f⋆adv]· n dΓf
, (6.42)
where uDf− and uDf+ are values of the discontinuous solution in elements Ωk and Ωk+1
located on either side of face f , and n is the outward pointing unit normal vector
associated with element Ωk (i.e. n = n−).
Now, equation (6.42) can be simplified by manipulating the term[G(uDf−)−G
(uDf+)]·
n. In particular, one may rewrite this term as follows
[G(uDf−)−G
(uDf+)]· n =
∫ uDf−
uDf+
(∂G
∂u(u)
)· n du. (6.43)
In accordance with the mean value theorem and the definition of G (equation (6.39)),
one may reformulate equation (6.43) as follows
[G(uDf−)−G
(uDf+)]· n =
((∂G
∂u(u)
∣∣∣∣ξ
)· n)(uDf− − uDf+
)
= (fadv (ξ) · n)(uDf− − uDf+
), (6.44)
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 146
where ξ ∈[min
(uDf−, u
Df+
),max
(uDf−, u
Df+
)]. On substituting equation (6.44) into
equation (6.42), one obtains
Ξadv =
Nf∑
f=1
∫
Γf
[(uDf+ − uDf−
)(f⋆adv − fadv (ξ)) · n
]dΓf
. (6.45)
Now, if f⋆adv is chosen to be an E-flux as defined by Osher in [75], one is assured that
(uDf+ − uDf−
)(f⋆adv − fadv (ξ)) · n ≤ 0, (6.46)
and therefore that
Ξadv =
Nf∑
f=1
∫
Γf
[(uDf+ − uDf−
)(f⋆adv − fadv (ξ)) · n
]dΓf
≤ 0. (6.47)
Note that a number of popular fluxes, including the aforementioned Lax-Friedrichs
advective numerical flux, can be categorized as E-fluxes. In order to demonstrate this
fact, consider defining the normal component of the Lax-Friedrichs flux as follows
f⋆adv · n = fadv · n+λ
2
(max
η∈[uDf−,uD
f+]
∣∣∣∣∂fadv∂u
(η) · n∣∣∣∣
)[[uD]] · n
=1
2
[(fadv
(uDf+)+ fadv
(uDf−))· n− λ
(uDf+ − uDf−
)(
maxη∈[uD
f−,uDf+]
∣∣∣∣∂fadv∂u
(η) · n∣∣∣∣
)].
(6.48)
On subtracting 12fadv (ξ)·n from equation (6.48) and multiplying the result by
(uDf+ − uDf−
),
one obtains
1
2
(uDf+ − uDf−
)(f⋆adv − fadv (ξ)) · n
=1
2
[ (uDf+ − uDf−
) (fadv
(uDf+)+ fadv
(uDf−))· n−
(uDf+ − uDf−
)fadv (ξ) · n
− λ(uDf+ − uDf−
)2(
maxη∈[uD
f−,uDf+]
∣∣∣∣∂fadv∂u
(η) · n∣∣∣∣
)]. (6.49)
The first two terms on the RHS cancel because of the mean value theorem and
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 147
therefore equation (6.49) becomes
1
2
(uDf+ − uDf−
)(f⋆adv − fadv (ξ))·n = −λ
2
(uDf+ − uDf−
)2(
maxη∈[uD
f−,uDf+]
∣∣∣∣∂fadv∂u
(η) · n∣∣∣∣
)≤ 0.
(6.50)
Thus, the Lax-Friedrichs flux is an E-flux (in accordance with equation (6.46)) and
for this choice of the advective numerical flux, Ξadv ≤ 0.
Now, having shown that a particular choice of f⋆adv ensures that Ξadv ≤ 0, one may
follow a similar approach and show that particular choices of u⋆ and f⋆dif ensure
that Ξdif ≤ 0. Towards this end, consider rewriting equation (6.35) in terms of a
summation over interfaces (Γf ) as follows
Ξdif =
Nf∑
f=1
∫
Γf
[ (uDf+ − uDf−
)f⋆dif + u⋆
(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))
+ uDf−fdif(uDf−,q
Df−
)− uDf+fdif
(uDf+,q
Df+
) ]· n dΓf
. (6.51)
Evidently, in order to ensure that Ξdif ≤ 0, one requires that the kernel of the integral
in equation (6.51) is non-positive, i.e.
[ (uDf+ − uDf−
)f⋆dif + u⋆
(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))
+ uDf−fdif(uDf−,q
Df−
)− uDf+fdif
(uDf+,q
Df+
) ]· n (6.52)
must be non-positive. There are a number of popular approaches for computing u⋆
and f⋆dif such that equation (6.52) is non-positive. In what follows, it will be shown
that the LDG approach is one such approach. For the LDG approach, u⋆ and f⋆dif · n
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 148
can be defined as follows
u⋆ = uD − β · [[uD]]
=1
2
(uDf+ + uDf−
)+(uDf+ − uDf−
)(β · n) , (6.53)
f⋆dif · n =(fdif+ τ [[uD]] + β[[fdif ]]
)· n
=1
2
(fdif(uDf+,q
Df+
)+ fdif
(uDf−,q
Df−
))· n
− τ(uDf+ − uDf−
)−(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))· n (β · n) . (6.54)
On substituting equations (6.53) and (6.54) into equation (6.52), one obtains
[ (uDf+ − uDf−
)(1
2
(fdif(uDf+,q
Df+
)+ fdif
(uDf−,q
Df−
))· n
− τ(uDf+ − uDf−
)−(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))· n (β · n)
)
+
(1
2
(uDf+ + uDf−
)+(uDf+ − uDf−
)(β · n)
)(fdif(uDf+,q
Df+
)− fdif
(uDf−,q
Df−
))· n
+ uDf−fdif(uDf−,q
Df−
)· n− uDf+fdif
(uDf+,q
Df+
)· n]
= −τ(uDf+ − uDf−
)2 ≤ 0. (6.55)
Thus, the LDG approach ensures that equation (6.52) is ≤ 0, and therefore that
Ξdif ≤ 0.
Finally, one may ensure that Ξalias ≤ 0 by utilizing exact L2 projections to construct
the fluxes fDk, adv and fDk, dif on Ωk and Γk, respectively. This will be shown via a brief
examination of these L2 projections.
Consider the exact L2 projection of a flux fDk on Ωk which is defined such that
∀i∫
Ωk
(fDk − f
(uDk))L3Dk,i dΩk = 0, (6.56)
where each L3Dk,i is a member of a 3D orthonormal polynomial basis of degree p − 1
on Ωk. Furthermore, the exact L2 projection of a flux fDk on Γk is defined such that
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 149
for each of the f = 1, . . . , 4 faces of the element boundary
∀l∫
Γf
(fDk − f
(uDk))· nf L
2Dk,f,l dΓf = 0, (6.57)
where each L2Dk,f,l is a member of a 2D orthonormal polynomial basis of degree p on
Γf .
Setting equations (6.56) and (6.57) aside for the moment, consider representing ∇uDkon Ωk and
(uDk − u⋆
)on Γf in terms of the orthonormal basis functions L3D
k,i and L2Dk,f,l
as follows
(∇uDk
)m=
Np∑
i=1
ζk,i,mL3Dk,i (x) , for m = 1, 2, 3, (6.58)
(uDk − u⋆
)f=
Nfp∑
l=1
ϕk,f,l L2Dk,f,l (x) , (6.59)
where ζk,i,m and ϕk,f,l are constant coefficients, and Np is the number of points required
to define a polynomial of degree p− 1 in 3D, i.e.
Np =p (p+ 1) (p+ 2)
6. (6.60)
Upon multiplying equation (6.56) by each coefficient ζk,i,m, substituting fadv in place
of f , and summing the result over m and i, one obtains
∫
Ωk
∇uDk ·(fDk, adv − fadv
(uDk))
dΩk = εΩk= 0. (6.61)
Similarly, on multiplying equation (6.57) by each coefficient ϕk,f,l, substituting fdif in
place of f , and summing the result over l and f , one obtains
∫
Γk
[(uDk − u⋆
) (fDk, dif − fdif
(uDk ,q
Dk
))]· n dΓk = εΓk
= 0. (6.62)
From equations (6.61), (6.62), and (6.36), it immediately follows that Ξalias = 0. This
completes the proof of Theorem 6.3.1.
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 150
Remark. There is at least one alternative approach for proving the stability of high-
order schemes for nonlinear advection-diffusion problems on unstructured meshes.
This approach (due to Cockburn and Shu [53]) utilizes the following definitions for
fdif and q
fdif =√b (u)q, q =
√b (u)∇u. (6.63)
In accordance with these definitions, the first order system in equations (6.3) and
(6.4) can be rewritten as follows
∂u
∂t+∇ ·
(fadv (u)−
√b (u)q
)= 0, (6.64)
q−√b (u)∇u = 0. (6.65)
It can be shown that if a DG scheme is employed to solve equations (6.64) and (6.65),
the stability of the scheme is governed by the following equation
1
2
N∑
k=1
(d
dt‖Uk‖2
M
)+
N∑
k=1
(QT
k
(B1/2k
)TMk B1/2
k Qk
)= Ξadv + Ξdif + Ξalias, (6.66)
where the quantity QTk
(B1/2k
)TMk B1/2
k Qk is a norm
‖Qk‖2(B1/2k
)TMk B
1/2k
= QTk
(B1/2k
)TMk B1/2
k Qk, (6.67)
because the matrix(B1/2k
)TMk B1/2
k is symmetric positive-definite, and where (as
before) Ξadv, Ξdif , and Ξalias are contributions from the advective flux, the diffusive
flux, and the aliasing error. As in Theorem 6.3.1, it can be shown that if the advective
numerical flux, common solution, and diffusive numerical flux are chosen appropri-
ately, one is assured that Ξadv ≤ 0 and Ξdif ≤ 0, and that if the discontinuous flux is
formed using exact L2 projections, one is assured that Ξalias ≤ 0.
The analysis above may appear to produce a stronger result than that of Theo-
rem 6.3.1, as there are no guidelines governing the mesh resolution or the degree of
nonlinearity of the flux. (Recall that Theorem 6.3.1 only guarantees stability in the
limit of infinite mesh resolution for fluxes for which b(uDk)is highly nonlinear, or
on coarser meshes for weakly nonlinear fluxes for which b(uDk)≈ bk). However, the
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 151
definitions of fdif and q used in this analysis (in equation (6.63)) cannot be general-
ized to handle nonlinear advection-diffusion problems that possess arbitrary diffusive
fluxes of the form fdif (u,∇u). In particular, it is unclear how the definitions in equa-
tion (6.63) can be applied to any diffusive flux that is not of the form fdif = −b (u)∇u.Conversely, it is relatively straightforward to rewrite an arbitrary nonlinear diffusive
flux fdif (u,∇u) utilizing the definitions of fdif and q in equations (6.3) and (6.4), as
one may simply form fdif = f (u,q) where q = ∇u. For this reason, the more general
approach (employed in Theorem 6.3.1) is the most frequently used in practice (as
noted in [76]), and is the primary approach that was utilized in this work.
Remark. One is frequently unable to compute exact L2 projections of high dimen-
sional or infinite dimensional nonlinear fluxes fadv(uDk)and fdif
(uDk ,q
Dk
)in problems
of practical interest. However, if numerical quadrature rules of sufficiently high-order
are employed in equations (6.56) and (6.57), the L2 projections can ensure that Ξalias
is of the order of machine zero. Nevertheless, in this case, the L2 projection proce-
dures will be substantially more expensive, from a computational standpoint, than
the collocation projection utilized previously to define fDk (in equation (4.8)).
It turns out that one may utilize collocation projection procedures to form the fluxes
while simultaneously reducing aliasing errors, if the solution points xi and flux points
xf,l are placed at the locations of quadrature points. This can be shown through a
careful examination of the aliasing errors εΩkand εΓk
. Towards this end, consider
utilizing equations (4.19) and (4.20) to transform εΩkand εΓk
from the physical space
to the reference space as follows
εΩk=εΩS
Jk=
1
Jk
∫
ΩS
∇uD ·(fDadv − fadv
(uD))
dΩS, (6.68)
εΓk=εΓS
Jk=
1
Jk
∫
ΓS
[(uD − u⋆
) (fDdif − fdif
(uD, qD
))]· n dΓS, (6.69)
where εΩSand εΓS
are measures of the aliasing errors in the reference space. (Note
that the transformation to reference space has been performed in order to simplify
the subsequent analysis).
Next, one may define ∇uD in equation (6.68) and(uD − u⋆
)in equation (6.69) as
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 152
follows
(∇uD
)m=
Np∑
i=1
ζi,mL3Di (x) , for m = 1, 2, 3, (6.70)
(uD − u⋆
)f=
Nfp∑
l=1
ϕf,l L2Df,l (x) , (6.71)
where ζi,m and ϕf,l are constant coefficients, L3Di is a member of the 3D orthonormal
polynomial basis of degree p − 1 inside the element (i.e. L3Di belongs to the class
of orthonormal basis functions that are of one degree lower than the basis functions
defined in Chapter 4), and L2Df,l is a member of the 2D orthonormal polynomial basis
of degree p on face f of the element (i.e. L2Df,l belongs to the class of orthonormal
basis functions that was defined in Chapter 3). Upon substituting equations (6.70)
and (6.71) into the expressions for εΩSand εΓS
in equations (6.68) and (6.69), one
obtains
εΩS=
3∑
m=1
Np∑
i=1
ζi,m
∫
ΩS
(fDm, adv − fm, adv
(uD))L3Di dΩS, (6.72)
εΓS=
4∑
f=1
Nfp∑
l=1
ϕf,l
∫
Γf
[(fDdif − fdif
(uD, qD
))· nf
]L2Df,l dΓf
=
3∑
m=1
4∑
f=1
Nfp∑
l=1
ϕf,l nf,m
∫
Γf
(fDm,f, dif − fm, dif
(uDf , q
Df
))L2Df,l dΓf , (6.73)
or equivalently,
εΩS=
3∑
m=1
Np∑
i=1
ζi,m (εΩS)i,m , (6.74)
εΓS=
3∑
m=1
4∑
f=1
Nfp∑
l=1
ϕf,l nf,m (εΓS)f,l,m , (6.75)
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 153
where
(εΩS)i,m =
∫
ΩS
(fDm, adv − fm, adv
(uD))L3Di dΩS, (6.76)
(εΓS)f,l,m =
∫
Γf
(fDm,f, dif − fm, dif
(uDf , q
Df
))L2Df,l dΓf . (6.77)
It should be noted that equations (6.76) and (6.77) are analogous to equations (6.56)
and (6.57) that define the L2 projections in physical space. In fact, equations (6.76)
and (6.77) simply define the component-wise L2 projections of the advective and
diffusive fluxes in reference space. Thus, if exact L2 projections of the fluxes are
performed then (εΩS)i,m and (εΓS
)f,l,m vanish as expected. However, for collocation
projections of the flux, this is not necessarily the case. In order to illustrate this
point, one may consider forming collocation projections of the fluxes as follows
fDm, adv =
Np∑
j=1
fm, adv
((uD)j)
ℓ 3Dj (x) =
Np∑
j=1
(fDm, adv
)jℓ 3Dj , (6.78)
fDm,f, dif =
Nfp∑
r=1
fm, dif
((uDf)r,(qDf
)r)ℓ 2Df,r (x) =
Nfp∑
r=1
(fDm,f, dif
)rℓ 2Df,r , (6.79)
where ℓ 3Dj is the 3D nodal polynomial of degree p that assumes the value of 1 at
solution j and the value of zero at all neighboring solution points (i.e. ℓ 3Dj belongs to
the class of nodal basis functions that was defined in Chapter 4), ℓ 2Df,r is the 2D nodal
polynomial of degree p that assumes the value of 1 at flux point f, r and the value of
zero at all neighboring flux points (i.e. ℓ 2Df,r belongs to the class of nodal basis functions
that was defined in Chapter 3), and where(fDm, adv
)jand
(fDm,f, dif
)rare the mth
components of the pointwise values of the advective and diffusive fluxes in reference
space ((fDadv
)jand
(fDf, dif
)r) that are related to the pointwise values of the fluxes in
physical space ((fDk, adv
)j= fk, adv
((uDk)j)
and(fDk,f, dif
)r= fk, dif
((uDf)r,(qDf
)r)) via
equation (4.15). Upon substituting equations (6.78) and (6.79) into equations (6.76)
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 154
and (6.77), one obtains
(εΩS)i,m =
Np∑
j=1
(fDm,adv
)j ∫
ΩS
ℓ 3Dj L3Di ΩS −
∫
ΩS
fm, adv
(uD)L3Di dΩS, (6.80)
(εΓS)f,l,m =
Nfp∑
r=1
(fDm,f, dif
)r ∫
Γf
ℓ 2Df,r L2Df,l dΓf −
∫
Γf
fm, dif
(uDf , q
Df
)L2Df,l dΓf . (6.81)
The terms ℓ 3Dj L3Di and ℓ 2Df,r L
2Df,l that appear in equations (6.80) and (6.81) are poly-
nomials of degree 2p− 1 in 3D and degree 2p in 2D, respectively. Thus, the integrals
of these terms can be computed exactly via quadrature rules of degree 2p− 1 in 3D
and degree 2p in 2D. More generally, a 3D quadrature rule (or equivalently a cubature
rule) of arbitrary degree can approximate the integral of ℓ 3Dj L3Di on ΩS as follows
∫
ΩS
ℓ 3Dj L3Di ΩS =
N3Dq∑
n=1
wn ℓ3Dj (ςn)L
3Di (ςn) + e3Dq , (6.82)
where N3Dq is the number of quadrature points, ςn’s are the point locations, wn’s are
the weights, and e3Dq is the quadrature error that vanishes for quadrature rules of
degree ≥ 2p−1. Similarly, a 2D quadrature rule of arbitrary degree can approximate
the integral of ℓ 2Df,r L2Df,l on Γf as follows
∫
Γf
ℓ 2Df,r L2Df,l dΓf =
N2Dq∑
s=1
ωs ℓ2Df,r (ϑs)L
2Df,l (ϑs) + e2Dq , (6.83)
where N2Dq is the number of quadrature points, ϑs’s are the point locations, ωs’s are
the weights, and e2Dq is the quadrature error that vanishes for quadrature rules of
degree ≥ 2p.
Upon substituting equations (6.82) and (6.83) into equations (6.80) and (6.81), one
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 155
obtains
(εΩS)i,m =
Np∑
j=1
(fDm, adv
)j N3Dq∑
n=1
wn ℓ3Dj (ςn)L
3Di (ςn)−
∫
ΩS
fm, adv
(uD)L3Di dΩS + E3D
q ,
(6.84)
(εΓS)f,l,m =
Nfp∑
r=1
(fDm,f, dif
)r N2Dq∑
s=1
ωs ℓ2Df,r (ϑs)L
2Df,l (ϑs)−
∫
Γf
fm,dif
(uDf , q
Df
)L2Df,l dΓf + E2D
q ,
(6.85)
where
E3Dq =
Np∑
j=1
(fDm, adv
)je3Dq , E2D
q =
Nfp∑
r=1
(fDm,f, dif
)re2Dq . (6.86)
Now, if the quadrature rules are required to have a particular number of points,
N3Dq = Np and N2D
q = Nfp, and the solution points xi (or equivalently xj) and flux
points xf,l (or equivalently xf,r) are placed at the locations of the quadrature points
ςn and ϑs, one obtains
(εΩS)i,m =
Np∑
j=1
fm,adv
(uD) ∣∣∣∣
ςj
Np∑
n=1
wn δjn L3Di (ςn)−
∫
ΩS
fm, adv
(uD)L3Di dΩS + E3D
q ,
(6.87)
(εΓS)f,l,m =
Nfp∑
r=1
fm,dif
(uDf , q
Df
) ∣∣∣∣ϑr
Nfp∑
s=1
ωs δr,s L2Df,l (ϑs)−
∫
Γf
fm, dif
(uDf , q
Df
)L2Df,l dΓf + E2D
q ,
(6.88)
or equivalently,
(εΩS)i,m =
Np∑
j=1
wj fm, adv
(uD) ∣∣∣∣
ςj
L3Di (ςj)−
∫
ΩS
fm, adv
(uD)L3Di dΩS + E3D
q , (6.89)
(εΓS)f,l,m =
Nfp∑
r=1
ωr fm, dif
(uDf , q
Df
) ∣∣∣∣ϑr
L2Df,l (ϑr)−
∫
Γf
fm, dif
(uDf , q
Df
)L2Df,l dΓf + E2D
q .
(6.90)
The sums in equations (6.89) and (6.90) act as numerical quadrature approximations
of the integral terms in each of the equations, and evidently, for quadrature rules
CHAPTER 6. ESFR FOR NONLINEAR ADV-DIFF ON TETRAHEDRA 156
of sufficient strength, the integral terms and their numerical approximations will
effectively cancel one another. In addition, (as mentioned previously) for quadrature
rules of order ≥ 2p − 1 in 3D and ≥ 2p in 2D, the terms E3Dq and E2D
q will vanish.
Thus, for quadrature rules of sufficient strength, (εΩS)i,m and (εΓS
)f,l,m approximately
vanish, εΩSand εΓS
approximately vanish, and in turn εΩkand εΓk
approximately
vanish. However, it is important to note that a quadrature rule of a very high degree
maybe required for this to occur, and that there will frequently not be enough solution
points or flux points to allow for this. Therefore, in practice, one should consider the
act of placing the solution points xi and flux points xf,l at the locations of quadrature
points as a procedure for reducing (but most likely not eliminating) aliasing errors.
This procedure results in an effective compromise between the extremes of eliminating
all aliasing errors via expensive L2 projections and creating large aliasing errors via
inexpensive collocation projections that employ solution and flux point locations that
differ from quadrature point locations.
Identifying high-order (or equivalently high-degree) quadrature rules with the correct
numbers of points N3Dq = Np and N2D
q = Nfp is not a trivial task. An effective
process for identifying these rules and thereby identifying desirable locations for the
solution points and flux points is discussed in the next chapter.
Chapter 7
Quadrature Points for Energy Sta-
ble Flux Reconstruction Schemes on
Triangles and Tetrahedra
157
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 158
7.1 Preamble
This chapter constructs new sets of quadrature rules on triangles and tetrahedra with
Np = (p+1)(p+2)/2 and Np = (p+1)(p+2)(p+3)/6 quadrature points, respectively.
When the quadrature point locations for these rules are utilized as solution and flux
point locations for the VCJH schemes, aliasing errors are minimized or eliminated (as
discussed in the previous chapter).
Note that this chapter is adapted from the article “Symmetric Quadrature Rules for
Simplexes Based on Sphere Closed Packed Lattice Arrangements” by D. M. Williams
and L. Shunn, which is pending submission to the Journal of Computational and
Applied Mathematics. The main results and over 95% of the text for this article were
contributed by D. M. Williams. An overview of this article is given below.
Sphere closed packed (SCP) lattice arrangements of points are well-suited for formu-
lating symmetric quadrature rules on simplexes, as they are symmetric under affine
transformations of the simplex unto itself in 2D and 3D. As a result, SCP lattice ar-
rangements have been utilized to formulate symmetric quadrature rules with Np = 1,
4, 10, 20, 35, and 56 points on the 3-simplex [77] (where it should be noted that the
points from these rules can exactly represent polynomials of order p = 0 to p = 5
on the 3-simplex). In what follows, the work on the 3-simplex is extended, and SCP
lattices are employed to identify symmetric quadrature rules with Np = 1, 3, 6, 10,
15, 21, 28, 36, 45, 55, and 66 points on the 2-simplex (where the points from these
rules can exactly represent polynomials of order p = 0 to p = 10 on the 2-simplex)
and Np = 84 points on the 3-simplex (where the points from this rule can exactly
represent a polynomial of up to order p = 6 on the 3-simplex). These rules are found
to be capable of exactly integrating polynomials of degree ≤ 17 in 2D and ≤ 9 in 3D.
7.2 Introduction
There have been significant efforts to identify high-order quadrature rules on d-simplex
and d-hypercube geometries as evidenced by the surveys in [78], [79], [80], [81], and
[82]. On the d-hypercube, it is well-known that optimal quadrature rules withNp = nd
points can be constructed from tensor products of the 1D Gauss-Legendre quadrature
rules with n points [83]. These quadrature rules are widely used because, in addition
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 159
to their optimality, they are symmetric (as they are invariant under reflections and
rotations that map the d-hypercube unto itself), possess positive weights, and have
points located within the interior of the d-hypercube. Due to these desirable prop-
erties, there have been attempts to extend the Gauss-Legendre rules to d-simplexes
(cf. [83] and [84]). The simplest of such approaches has involved constructing Gauss-
Legendre rules on the d-hypercube and then degenerating vertices until the d-simplex
is obtained. However, in general the resulting rules are no longer symmetric on the
d-simplex, as they contain anisotropic clusters of points near the degenerate vertices.
In addition, the optimality of these rules has yet to be shown analytically. In fact, to
the author’s knowledge, no one has identified a family of symmetric quadrature rules
on the d-simplex for which optimality can be rigorously proven. For this reason, the
formulation of quadrature rules on the d-simplex remains an open area of research, as
demonstrated by the recent work presented in [85], [86], [87], [88], [89], [90], and [77].
Of particular interest, is the effort by Shunn and Ham in [77] to construct quadrature
rules on the 3-simplex based on cubic close packing (CCP) arrangements of points.
In their work, the CCP arrangements of points (i.e. the CCP lattices) are defined by
the centers of spheres in the CCP configuration, as shown in Figure (7.1) for the cases
of Np = 1, 4, 10, 20, 35, 56, and 84.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 160
(a) Np = 1 (b) Np = 4 (c) Np = 10
(d) Np = 20 (e) Np = 35 (f) Np = 56
(g) Np = 84
Figure 7.1: Cubic closed packed (CCP) configurations on tetrahedra with Np = 1, 4, 10,20, 35, 56, and 84 points.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 161
The CCP lattices are viewed as the 3-simplex analog to the uniform cartesian lattices
on which the Gauss-Legendre rules on the 3-hypercube are based, in the sense that
they possess the same properties of symmetry on the 3-simplex as uniform cartesian
lattices possess on the 3-hypercube. Based on this fact, it was supposed that an
optimal family of symmetric quadrature rules on the 3-simplex could be obtained
by employing the CCP lattices as initial conditions to optimization procedures for
identifying the rules. Following this approach, a family of symmetric, locally optimal
quadrature rules on the 3-simplex with Np = 1, 4, 10, 20, 35, and 56 points was
obtained [77].
This work attempts to extend the approach in [77] to identify new quadrature rules
on the d-simplex for the cases of d = 2 and d = 3. This extension requires the
construction of CCP lattices in 2-space (sometimes referred to as ‘hexagonal packing’
lattices), which are shown in Figure (7.2) for the cases of Np = 1, 3, 6, 10, 15, 21, 28,
36, 45, 55, and 66.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 162
(a) Np = 1 (b) Np = 3 (c) Np = 6
(d) Np = 10 (e) Np = 15 (f) Np = 21
(g) Np = 28 (h) Np = 36 (i) Np = 45
(j) Np = 55 (k) Np = 66
Figure 7.2: Cubic closed packed (CCP) configurations on triangles with Np = 1, 3, 6, 10,15, 21, 28, 36, 45, 55, and 66 points.
For convenience, the analogs of the CCP lattices in d-space will henceforth be referred
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 163
to as d-sphere close packed lattices (d-SCP lattices). It is useful to note that, in
general, the number of points in the d-SCP lattices (the values of Np for the lattices)
can be expressed as a function of Nl, the number of layers of d-spheres in the lattices,
as follows
Np =(Nl + 1)!
d! (Nl + 1− d)! , (7.1)
where each layer is of dimension d− 1.
The remainder of this chapter is structured as follows. Section 3 presents requirements
for the quadrature rules and introduces the theoretical machinery that enables these
requirements to be enforced. Section 4 discusses the practical procedure for obtaining
the quadrature rules through solving nonlinearly constrained optimization problems.
Finally, section 5 and Appendices B and C summarize the quadrature point locations
and weights for the resulting rules.
7.3 Overview of the Theory of Quadrature Rules
on Simplexes
This section will describe the theoretical basis of the standard approach for finding
quadrature rules on d-simplexes. The description presented here is a generalization
of the description of Shunn and Ham in [77].
7.3.1 Preliminary Definition of the Problem
Consider the domain Ω which resides in a d-dimensional space with coordinate vector
x ∈ Rd. Suppose that a scalar-valued function f(x) is well-defined on Ω (such that
f(x) ∈ R ∀x ∈ Ω) and that the integral of f(x) on Ω is also well-defined such that
F ≡∫
Ω
f(x) dΩ, F ∈ R. (7.2)
In this case, F can be approximated using a quadrature rule (F) as follows
F ≡ V (Ω)
[Np∑
i=1
wi f (xi)
]≈ F, (7.3)
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 164
where V (Ω) denotes the volume of Ω, and xi and wi denote the locations and weights
of the quadrature points, respectively. Note that the error ǫ introduced by the quadra-
ture approximation in equation (7.3) can be defined as follows
ǫ ≡ F − FV (Ω)
. (7.4)
Now, suppose that Ω is defined to be the d-simplex with vertices v1, . . . ,vd+1. One
may redefine the quadrature rule in equation (7.3) in terms of the d-simplicial def-
initions of the volume V (Ω) and the quadrature point locations xi. In particular,
the volume of the d-simplex can be defined using the Cayley-Menger determinant as
follows
V (Ω) =1
d!det (B) , (7.5)
where B is a matrix which has columns that are constructed from the differences
between vertices of the d-simplex, i.e.
B =(v2 − v1) · · · (vd+1 − v1)
. (7.6)
In addition, the quadrature point locations xi can be expressed as the following linear
combinations of the d-simplex vertices
xi =d+1∑
j=1
ai,jvj. (7.7)
Upon substituting the expressions for V (Ω) and xi from equations (7.5) and (7.7)
into equation (7.3), one obtains the following general expression for the quadrature
rule F on the d-simplex
F =1
d!det (B)
[Np∑
i=1
wi f
(d+1∑
j=1
ai,jvj
)]. (7.8)
7.3.2 Quadrature Rule Requirements
Before proceeding further, it is useful to review the requirements that quadrature
rules are usually required to satisfy. In accordance with the discussions in [80], each
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 165
quadrature rule is required to:
• Integrate a polynomial of the highest possible order and minimize the magnitude
of the truncation error term.
• Ensure that all quadrature weights are non-negative in order to minimize the
possibility of cancellation errors.
• Ensure that there are no quadrature points located outside of the d-simplex.
• Ensure that all quadrature points are symmetrically arranged within the d-
simplex. The quadrature rule must be invariant under affine transformations of
the d-simplex unto itself.
In addition, on the 2-simplex it is desirable (but not required) that all quadrature
rules preserve the layered structure of the corresponding 2-SCP configuration. More
precisely, it is convenient if the quadrature rules have rows of points for which the
number of points is the same as the number of points in a complementary row of
the 2-SCP configuration. Here it is important to note that a row of points is defined
such that the y coordinates of all the points in a given row are greater than all y
coordinates of the points in the row below and less than all y coordinates of points
in the row above (with natural exceptions for the top-most and bottom-most rows).
When the points can be partitioned into rows in this way, it facilitates the creation
of mappings between the indices of quadrature points on pairs of 2-simplexes. These
mappings are useful in certain applications, for example, in numerical integration
procedures on triangular (2-simplex) faces of elements in 3D meshes of tetrahedra
(3-simplexes) that are frequently employed in discontinuous Finite Element methods.
Refer to Appendix B for details regarding this topic.
7.3.3 Enforcement of Quadrature Rule Requirements
Evidently, the objective is to find the particular rule (or set of rules) that satisfies
the requirements of the previous section. Of these requirements, it is perhaps most
important to find a quadrature rule which satisfies the first one, namely finding a
quadrature rule which integrates a polynomial of the highest possible degree and
minimizes the approximation error ǫ (equation (7.4)) for a given number of quadrature
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 166
points Np. More precisely, ǫ ∼ O (δn) must be minimized, where n is the highest
possible order, and where δ is the ‘characteristic’ length of edges between neighboring
vertices vk and vk+1 of the d-simplex (where k ≤ d). In order to begin finding a
quadrature rule that meets this criterion, consider approximating f (x) with a nth
order Taylor series expansion about the centroid x0 of Ω (denoted by fn (x)) where
fn (x) ≡ f (x) +O (δn) , (7.9)
so that equation (7.8) becomes
F =1
d!det (B)
[Np∑
i=1
wi fn
(d+1∑
j=1
ai,jvj
)]+ O
(δn+d
). (7.10)
Setting equation (7.10) aside for the moment, consider integrating both sides of equa-
tion (7.9) over the domain Ω in order to obtain the following expression for F
F =
∫
Ω
fn (x) dΩ+O(δn+d
)≡ Fn +O
(δn+d
). (7.11)
Upon substituting F and F from equations (7.10) and (7.11), and V (Ω) from equa-
tion (7.5) into equation (7.4), one obtains the following expression for the error term ǫ
ǫ =Fn d!
det (B)−[
Np∑
i=1
wi fn
(d+1∑
j=1
ai,jvj
)]+O (δn) ,
(7.12)
which is equivalent to
ǫ = ǫn +O (δn) , (7.13)
where
ǫn ≡Fn d!
det (B)−[
Np∑
i=1
wi fn
(d+1∑
j=1
ai,jvj
)]. (7.14)
It is well-known that ǫn is proportional to δn [77]. Therefore, upon combining this
result and the result of equation (7.13), one concludes that
ǫ ∼ O (δn) , (7.15)
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 167
as expected.
From this analysis, it can be seen that the quadrature rule error scales with the order
of the term ǫn. Therefore, satisfying the first quadrature rule requirement simplifies
to optimizing the quadrature point locations and weights such that ǫn is minimized
for the maximum possible value of n.
The remaining quadrature rule requirements can be satisfied as follows. The sec-
ond requirement states that the quadrature weights must be non-negative, which is
equivalent to requiring that
wi ≥ 0. (7.16)
In addition, the third requirement states that the quadrature points must not lie
outside of the d-simplex, which is equivalent to placing the following conditions on
the coefficients ai,j
ai,j ≥ 0,
d+1∑
j=1
ai,j = 1. (7.17)
The fourth requirement states that the quadrature points must be arranged in a sym-
metric fashion within the d-simplex. In order to satisfy this requirement, one must
enforce additional constraints on the coefficients ai,j. Prior to formulating these con-
straints, it is convenient to first formulate a symmetric d-simplex on which to enforce
them. Of course, symmetry constraints may be enforced on any d-simplex, as a linear
mapping to transform quadrature points from the initial simplex to any other sim-
plex of the same dimension can always be defined. However, it is more convenient to
construct these constraints on the standard, equilateral d-simplex (denoted by ΩS)
whose centroid (x0) is located at the origin. Evidently, ΩS is convenient to utilize
because it has d!-fold symmetry in the sense that all of the d! unique, affine, sym-
metric transformations that are possible for the d-simplex merely result in ΩS being
transformed unto itself. However, it also possesses some additional useful properties.
In particular, the ‘characteristic’ edge length δ is straightforward to define, as the
length of each edge is identical. Furthermore, because x0 is located at the origin,
one is ensured that Taylor series expansions about x0 are simple to formulate, as all
terms which contain components of the vector (x− x0)m can be reduced to contain
only components of xm.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 168
Now, having established the useful properties of the equilateral d-simplex ΩS, it is
important to examine the analytical form for the equilateral 2-simplex and 3-simplex
that will serve as the focus for the remainder of this chapter. The equilateral 2-simplex
with edge length δ is formed from the following three vertices
v1 = δ
(−12,−√3
6
)
v2 = δ
(1
2,−√3
6
)
v3 = δ
(0,
√3
3
). (7.18)
Upon substituting the vertices from equation (7.18) into equation (7.5), one finds that
the 2-simplex has a volume of V (ΩS) = δ2√3/4. In addition, it is useful to note that
equation (7.2) can be reformulated on the 2-simplex as follows
F =
∫ y1
y0
∫ x1
x0
f (x, y)dx dy, (7.19)
where the limits of integration x0, x1, y0, and y1 are defined as follows
x0 = −√3
3
(√3
3δ − y
)x1 =
√3
3
(√3
3δ − y
)(7.20)
y0 = −√3
6δ y1 =
√3
3δ. (7.21)
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 169
The equilateral 3-simplex with edge length δ is formed from the following four vertices
v1 = δ
(−12,−√3
6,−√6
12
)
v2 = δ
(1
2,−√3
6,−√6
12
)
v3 = δ
(0,
√3
3,−√6
12
)
v4 = δ
(0, 0,
√6
4
). (7.22)
Upon substituting the vertices from equation (7.22) into equation (7.5), one finds that
the 3-simplex has a volume of V (ΩS) = δ3√2/12. In addition, it is useful to note
that equation (7.2) can be reformulated on the 3-simplex as follows
F =
∫ z1
z0
∫ y1
y0
∫ x1
x0
f (x, y) dx dy dz, (7.23)
where the limits of integration x0, x1, y0, y1, z0, and z1 are defined as follows
x0 = −√3
3
[√2
2
(√6
4δ − z
)− y]
x1 =
√3
3
[√2
2
(√6
4δ − z
)− y]
(7.24)
y0 = −√2
4
(√6
4δ − z
)y1 =
√2
2
(√6
4δ − z
)(7.25)
z0 = −√6
12δ z1 =
√6
4δ. (7.26)
Having established precise definitions for the d = 2 and d = 3 standard simplexes,
symmetry constraints on the coefficients ai,j for each of these simplexes may now
be formulated. For convenience, one may impose the constraints by introducing pa-
rameters αk which symmetrically parameterize the point locations xi, and indirectly
via equation (7.7) parameterize (and constrain) the ai,j’s. In this way, the problem
simplifies to finding the symmetry parameters αk, from which xi (αk) and ai,j (xi)
immediately follow. This is convenient because there are frequently far fewer pa-
rameters αk than there are coefficients ai,j. For example, consider the symmetric
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 170
parameterization of the 2-simplex with Np = 15 points illustrated in Figure (7.3).
Figure 7.3: Np = 15 points symmetrically arranged on the 2-simplex (triangle).
In this case, there are a total of 15 (d+ 1) = 45 unknown coefficients ai,j, whereas
there are only five unknown parameters αk (where k = 1, . . . , 5). The reduction in the
number of variables arises naturally from the symmetry requirements. In particular,
due to symmetry, six of the fifteen points are required to lie along lines between
the centroid x0 and the vertices v1, v2, and v3 of the 2-simplex. Because there are
three vertices, there are a total of two points along lines between each vertex and the
centroid, and there are in turn two symmetry parameters α1 and α2 which define the
point locations xi as follows
xi = α1vi where i = 1, 2, 3
xi = α2vi−3 where i = 4, 5, 6. (7.27)
Furthermore, there are three points which are required to lie along lines between the
centroid and the midpoints of the simplex edges. The locations of these points can
be defined using a single symmetry parameter α3 as follows
xi = α3 (vj + vm) where i = 7, 8, 9, 1 ≤ j,m ≤ 3, j 6= m. (7.28)
Finally, there are six points which must lie along lines which emanate from the cen-
troid and intersect the simplex edges at locations between the edge midpoints and the
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 171
vertices. The locations of these points can be defined using two symmetry parameters
α4 and α5 as follows
xi = α4vj + α5vm where i = 10, 11, 12
xi = α5vj + α4vm where i = 13, 14, 15. (7.29)
Equations (7.27) - (7.29) provide a complete description of the symmetric point lo-
cations in terms of the five symmetry variables α1, . . . , α5.
Next, in order to recover the coefficients ai,j from the parameters αk, one may refor-
mulate equation (7.7) in terms of the following linear system
(v1
1
)(v2
1
)(v3
1
)ai,1ai,2ai,3
=
xiyi1
, (7.30)
where the fact that ai,1 + ai,2 + ai,3 = 1 (equation (7.17)) has been used.
A similar procedure for enforcing the symmetry constraints can be employed on the
3-simplex. For example, consider the symmetric arrangement of Np = 84 points on
the 3-simplex illustrated in Figure (7.4).
Figure 7.4: Np = 84 points symmetrically arranged on the 3-simplex (tetrahedron).
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 172
There are a total of 84 (d+ 1) = 336 parameters ai,j, however, it turns out that
symmetry constraints need only be enforced on fourteen parameters αk (where k =
1, . . . , 14). In particular, due to symmetry, eight of the eighty-four points must be
located along lines from the centroid x0 to each of the vertices v1, v2, v3, and v4.
The locations of these points can be expressed in terms of the symmetry parameters
α1 and α2 as follows
xi = α1vi where i = 1, 2, 3, 4
xi = α2vi−4 where i = 5, 6, 7, 8. (7.31)
In addition, twelve points must located along lines between the centroid and the
midpoints of the simplex edges. The locations of these points can be expressed in
terms of the symmetry parameters α3 and α4 as follows
xi = α3 (vj + vm) where i = 9, . . . , 14 1 ≤ j,m ≤ 4, j 6= m
xi = α4 (vj + vm) where i = 15, . . . , 20. (7.32)
Furthermore, twenty-four points must be located along lines which emanate from the
centroid and intersect the simplex edges at locations between the edge midpoints and
the vertices. The locations of these points can be expressed in terms of the symmetry
parameters α5, α6, α7, and α8 as follows
xi = α5vj + α6vm where i = 21, . . . , 26
xi = α6vj + α5vm where i = 27, . . . , 32
xi = α7vj + α8vm where i = 33, . . . , 38
xi = α8vj + α7vm where i = 39, . . . , 44. (7.33)
Finally, the locations of the remaining forty points are composed from all possible
unique, symmetric, linear combinations of the three vertices of each face. The loca-
tions of these points can be expressed in terms of the symmetry parameters α9, . . . , α14
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 173
as follows
xi = α9 (vj + vm + vl) where i = 45, 46, 47, 48
1 ≤ j,m, l ≤ 4, j 6= m 6= l
xi = α10vj + α11 (vm + vl) where i = 49, 50, 51, 52
xi = α10vl + α11 (vj + vm) where i = 53, 54, 55, 56
xi = α10vm + α11 (vl + vj) where i = 57, 58, 59, 60
xi = α12vj + α13vm + α14vl where i = 61, 62, 63, 64
xi = α12vl + α13vj + α14vm where i = 65, 66, 67, 68
xi = α12vm + α13vl + α14vj where i = 69, 70, 71, 72
xi = α12vj + α14vm + α13vl where i = 73, 74, 75, 76
xi = α12vl + α14vj + α13vm where i = 77, 78, 79, 80
xi = α12vm + α14vl + α13vj where i = 81, 82, 83, 84. (7.34)
Equations (7.31) - (7.34) provide a complete description of the symmetric point lo-
cations in terms of the fourteen symmetry variables α1, . . . , α14.
One may recover the coefficients ai,j from the parameters αk by solving the following
linear system of equations
(v1
1
)(v2
1
)(v3
1
)(v4
1
)
ai,1ai,2ai,3ai,4
=
xiyizi1
, (7.35)
where the fact that ai,1 + ai,2 + ai,3 + ai,4 = 1 (equation (7.17)) has been used.
Finally, the fifth requirement (which is a soft requirement) states that the quadrature
points on the 2-simplex should be arranged in a way that is consistent with the 2-
SCP layering. Since this is not a strict requirement, it need not be enforced directly.
However, it can be encouraged by employing 2-SCP configurations (or perturbed
2-SCP configurations) as initial conditions for the optimization process utilized in
identifying the quadrature rules. This process will be discussed in more detail in the
next section.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 174
7.4 Computation of Optimal Quadrature Rules
Quadrature rules on the 2 and 3-simplexes can be obtained by solving a series of
constrained optimization problems. In each of these problems, the symmetry variables
(αk’s) and the quadrature weights (wi’s) are treated as unknowns. These unknowns
are subject to the inequality and equality constraints put forth in section 7.3.3, which,
for convenience, can be reformulated as follows.
7.4.1 Reformulating the Inequality and Equality Constraints
Inequality constraints on the unknowns are obtained from equations (7.16) and (7.17).
The constraints in equation (7.16) apply directly to the wi’s, however the constraints
in equation (7.17) apply to the ai,j’s and cannot be easily applied to the αk’s. Nev-
ertheless, in practice, it is frequently sufficient to weakly enforce the constraints in
equation (7.17) by enforcing the following constraints on the αk’s
1 ≥ αk ≥ 0. (7.36)
These constraints do not ensure that the quadrature points remain inside of the d-
simplex, but they encourage this by ensuring that all points lie inside of the d-sphere
of radius d|v1| centered at the centroid (x0) of the d-simplex.
Equality constraints on the unknowns are obtained by evaluating ǫn in equation (7.14).
In particular, in order to obtain a quadrature rule with truncation error of order
δn, it is required that all error terms of order less than δn vanish, and therefore,
equality constraints are obtained by insisting that all error terms ǫm vanish for m =
0, . . . , (n− 1). The requirement that ǫm = 0 can be written in matrix form as follows
ǫm = δm (∂fm)T Cm um = 0, (7.37)
where (∂fm)T is a vector of partial derivative terms of order m, Cm is a matrix
of constant coefficients (which need not be full rank), and um is a vector containing
monomial terms of the unknown αk’s and wi’s. In order to ensure that equation (7.37)
holds, one requires that
Cm um = 0, (7.38)
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 175
or equivalently
Cm um = 0, (7.39)
where Cm is the matrix that arises from reducingCm to row echelon form and removing
all rows of zeros. Evidently rank (Cm) = rank (Cm) if and only if Cm is full rank.
Now, the matrices Cm for m = 0, . . . , (n− 1) can be combined together into the
matrix C, and similarly the vectors um can be combined together in order to form the
vector u. Thus, the final nonlinear system of equality constraints for the quadrature
rule of order n becomes Cu = 0.
Evidently, the number of rows in C is equivalent to the number of equality constraints
that must be satisfied. In order to ensure that it is possible for the system of equations
Cu = 0 to have a solution, it is necessary for the number of unknowns (αk’s and wi’s)
to equal or exceed the number of equations. Therefore, for each of the quadrature rules
obtained in this work, the value of n was chosen such that the number of unknowns
was greater than or equal to the number of unique equality constraints associated
with the error terms of order less than n.
Having established a methodology for forming the constraints and identifying the
order of a quadrature rule n, one may proceed to form an objective function and
solve the resulting optimization problem.
7.4.2 Forming the Objective Function and Solving the Opti-
mization Problem
In following the approach of [77], the objective function Jn can be formed as the sum
of the squares of all unique equality constraints of degree n, i.e.
Jn = uTnCTn Cn un. (7.40)
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 176
In accordance with this definition, the constrained optimization problem for each
quadrature rule of degree n becomes
minimize Jn = uTnCTn Cn un
subject to Cu = 0
wi ≥ 0, 1 ≥ αk ≥ 0. (7.41)
In order to obtain the quadrature rules in this work, equation (7.41) was solved iter-
atively using a sequential quadratic programming (SQP) algorithm. This algorithm
was chosen due to its ability to treat equality and inequality constraints of linear and
nonlinear type. In accordance with the standard SQP approach (described in [91],
[92], [93], and [94]), all constraints were incorporated into a Lagrangian function, and
approximate Hessians of this function (computed via the Broyden-Fletcher-Goldfarb-
Shanno (BFGS) method [95], [96], [97], and [98]) were used to create a sequence of
quadratic programming subproblems. The subproblems were solved using an Active-
Set strategy (described in [99] and [100]) in order to obtain search directions, and
a line search algorithm (utilizing a merit function described in [101] and [102]) was
employed in order to identify step lengths. In turn, these step lengths and search
directions were used to successively update the solution.
The converged, optimal solutions were sensitive to initial conditions, indicating that
the optimization problems possessed multiple local minima. In an effort to obtain
the global minima, 106 initial conditions were employed during the optimization of
each quadrature rule with Np = 3, 6, 10, 15, 21, and 28 points on the 2-simplex,
105 initial conditions were employed for the quadrature rules with Np = 36, 45, 55,
and 66 points on the 2-simplex, and 2.5 × 104 initial conditions were employed for
the quadrature rule with Np = 84 points on the 3-simplex. These initial conditions
were obtained by subjecting the d-SCP configurations to random perturbations. The
optimal quadrature rules obtained through this process are discussed in the next
section.
7.5 Results
Table (C.1) in Appendix C contains a set of optimal quadrature rules on the 2-simplex
with Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. Note that the quadrature
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 177
rule parameters in the table are given to fifteen decimal places, in accordance with
the limits on the precision of the computations. Figures (7.5)–(7.7) illustrate the
arrangements of points for each of the rules. From the figures, it is apparent that the
points are arranged symmetrically on the 2-simplex, the points lie within the interior
of the 2-simplex, and all point arrangements can be partitioned into rows, and are
thus consistent with the corresponding 2-SCP configurations.
(a) Np = 1 (b) Np = 3
(c) Np = 6 (d) Np = 10
Figure 7.5: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 1, 3, 6, and 10 points for the triangle (the 2-simplex). The size of each point is scaledby the absolute value of the logarithm of the associated quadrature weight.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 178
(a) Np = 15 (b) Np = 21
(c) Np = 28 (d) Np = 36
Figure 7.6: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 15, 21, 28, and 36 points for the triangle (the 2-simplex). The size of each point isscaled by the absolute value of the logarithm of the associated quadrature weight.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 179
(a) Np = 45 (b) Np = 55
(c) Np = 66
Figure 7.7: Quadrature point locations for the quadrature rules with 2-SCP structure andNp = 45, 55, and 66 points for the triangle (the 2-simplex). The size of each point is scaledby the absolute value of the logarithm of the associated quadrature weight.
For the sake of completeness, the requirement on consistency with the 2-SCP config-
urations was relaxed, and an alternate set of quadrature rules with Np = 45, 55, and
66 points was obtained on the 2-simplex. These quadrature rules (summarized by Ta-
ble (C.2) and Figure (7.8)) are more optimal than their counterparts in Table (C.1),
as they produce roughly an additional order of magnitude decrease in the objective
function.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 180
(a) Np = 45 (b) Np = 55
(c) Np = 66
Figure 7.8: Quadrature point locations for the quadrature rules with non-(2-SCP) structureand Np = 45, 55, and 66 points for the triangle (the 2-simplex). The size of each point isscaled by the absolute value of the logarithm of the associated quadrature weight.
Nevertheless, both sets of quadrature rules produce the same order of truncation error
(cf. Table (7.1)), as the magnitude of the objective function only governs the size of
the constant that multiplies the truncation error term, and not the order of the term
itself. Recall that the order of the truncation error term is determined by the number
of degrees of freedom and the number of nonlinear constraints, which are identical
Rules that are inconsistent with 2-SCP configurationsNl Np Order9 45 1510 55 1611 66 18
Table 7.1: Order of accuracy estimates for quadrature rules on the 2-simplex with Np = 1,3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points. Estimates are provided for quadrature rulesthat are consistent and inconsistent with the 2-SCP configurations.
Based on the truncation errors in Table (7.1), the quadrature rules in Tables (C.1)
and (C.2) are capable of exactly integrating all polynomials of degree 0 ≤ p ≤ 17 on
the 2-simplex.
Finally, Table (C.3) contains the optimal quadrature rule with Np = 84 points on the
3-simplex. Figure (7.9) illustrates this rule.
CHAPTER 7. QUADRATURE POINTS FOR ESFR SCHEMES 182
Figure 7.9: Quadrature point locations for the quadrature rule with Np = 84 points for thetetrahedron (the 3-simplex). The size of each point is scaled by the associated quadratureweight.
The truncation error for this rule was found to be O(δ10), enabling it to exactly
integrate polynomials of degree ≤ 9 on the 3-simplex.
Part III
Nonlinear Numerical Experiments
183
Chapter 8
Governing Equations
184
CHAPTER 8. GOVERNING EQUATIONS 185
8.1 Preamble
In order to evaluate the nonlinear behavior of the VCJH schemes, they were employed
to solve the nonlinear Navier-Stokes (NS) equations on 2D and 3D grids of triangular
and tetrahedral elements. In order to help ensure the stability of the schemes, the
flux points and solution points for the VCJH schemes on each grid were placed at the
locations of the SCP quadrature points from the previous chapter, unless otherwise
indicated.
Before proceeding further, it should be noted that the NS equations in 2D and 3D
are of particular interest because they have the capability to accurately model the
flows of Newtonian fluids that arise in many real-world applications. In addition,
the NS equations are (in some sense) a natural extension of the linear and nonlin-
ear advection-diffusion equations that were used in previous chapters to develop the
theoretical foundation of the VCJH schemes.
8.2 Definitions of the Navier-Stokes Equations
The NS equations in 2D and 3D can be written as follows
∂U
∂t+∇ · F(U,∇U) = 0, (8.1)
where U represents the conserved variables (which are scalars) and F represents
the flux vector that is composed from inviscid and viscous parts: F = Finv (U) −Fvisc (U,∇U). In 2D, the conserved variables are defined as follows
U =
ρρuρvE
, (8.2)
where ρ = ρ (x, y, t) is the density, u = u (x, y, t) and v = v (x, y, t) are the velocity
components, E = p/ (γ − 1) + (1/2)ρ (u2 + v2) is the total energy, p = p (x, y, t) is
the pressure, and γ is the ratio of specific heats. In addition, the inviscid and viscous
fluxes in 2D can be defined in terms of their components along the x and y coordinate
directions, i.e. Finv = (finv, ginv) and Fvisc = (fvisc, gvisc). Here, the inviscid flux
CHAPTER 8. GOVERNING EQUATIONS 186
components are defined such that
finv =
ρuρu2 + pρuv
u(E + p)
, ginv =
ρvρuv
ρv2 + pv(E + p)
, (8.3)
and the viscous flux components are defined such that
fvisc = µ
02ux + λ(ux + vy)
vx + uyu[2ux + λ(ux + vy)] + v(vx + uy) +
Cp
PrTx
,
gvisc = µ
0vx + uy
2vy + λ(ux + vy)
v[2vy + λ(ux + vy)] + u(vx + uy) +Cp
PrTy
, (8.4)
where µ is the dynamic viscosity, λ is the bulk viscosity coefficient, T = p/ (ρR) is
the temperature, R is the gas constant, Cp is the specific heat capacity at constant
pressure, and Pr is the Prandtl number. It should be noted that the terms with
subscripts x and y in equation (8.4) signify first derivatives in x and y (for example
Ty =∂T∂y).
In 3D, the conserved variables U are defined as follows
U =
ρρuρvρwE
, (8.5)
where ρ = ρ (x, y, z, t), u = u (x, y, z, t), v = v (x, y, z, t), w = w (x, y, z, t) (where w
is the velocity component in the z direction), E = p/ (γ − 1)+ (1/2)ρ (u2 + v2 + w2),
and p = p (x, y, z, t). In addition, the inviscid and viscous fluxes in 3D can be
defined in terms of their components along the coordinate directions, i.e. Finv =
CHAPTER 8. GOVERNING EQUATIONS 187
(finv, ginv, hinv) and Fvisc = (fvisc, gvisc, hvisc), where
Before proceeding further, it is useful to rewrite the 2D and 3D versions of the NS
equations in non-dimensional form. Non-dimensionalization reduces the round-off
errors associated with numerical computations by ensuring that certain critical quan-
tities (e.g. the density, the velocities, the spatial coordinates, and the temperature)
are normalized so that they are O (1). In order to facilitate the computation of non-
dimensional quantities, it is necessary to define dimensional reference values of the
density (ρ∞), the flow speed (u∞), the characteristic length scale (L), and the tem-
perature (T∞). Using these quantities, the following additional dimensional reference
quantities can be computed
t∞ =L
u∞, p∞ = ρ∞u
2∞, µ∞ = ρ∞u∞L. (8.8)
CHAPTER 8. GOVERNING EQUATIONS 188
Now that these quantities are specified, it is straightforward to construct non-dimensional
forms of all the quantities in the NS equations as follows
ρ⋆ =ρ
ρ∞, t⋆ =
t
t∞, x⋆ =
x
L, y⋆ =
y
L, z⋆ =
z
L, (8.9)
T ⋆ =T
T∞, u⋆ =
u
u∞, v⋆ =
v
u∞, w⋆ =
w
u∞, (8.10)
p⋆ =p
p∞= ρ⋆R⋆T ⋆, R⋆ =
RT∞u2∞
, µ⋆ =µ
µ∞
, (8.11)
E⋆ =p⋆
γ − 1+
1
2ρ⋆(u⋆
2
+ v⋆2
+ w⋆2), C⋆
p =CpT∞u2∞
, (8.12)
where each non-dimensional quantity is denoted with a (⋆) superscript. Upon substi-
tuting equations (8.9) – (8.12) into the expressions for the conservative variables and
the flux vectors, one obtains a set of non-dimensional conservative variables denoted
by U⋆, and a set of non-dimensional fluxes denoted by F⋆. Utilizing these quantities,
the non-dimensional NS equations can be written as follows
∂U⋆
∂t⋆+∇⋆ · F⋆(U⋆,∇⋆U⋆) = 0, (8.13)
where ∇⋆ =(
∂∂x⋆ ,
∂∂y⋆, ∂∂z⋆
)is the non-dimensional form of the gradient operator.
It should be noted that the behavior of solutions to equation (8.13) is (to a large
extent) dictated by the Prandtl number Pr (discussed previously), and the Reynolds
(Re) and Mach (M) numbers, which are non-dimensional quantities that are defined
as follows
Re =ρ∞u∞L
µ∞, M =
u∞√γRT∞
. (8.14)
Finally, it should be noted that equation (8.13) contains second derivatives of the non-
dimensional conservative variables and can therefore be classified as a ‘2nd-order’
system of PDEs. However, this system can easily be reformulated as a first-order
system by following the approach of sections 3.2, 4.2, and 6.2, i.e. by eliminating∇⋆U⋆
from equation (8.13) and replacing it with the non-dimensional auxiliary variable
CHAPTER 8. GOVERNING EQUATIONS 189
denoted by Q⋆ as follows
∂U⋆
∂t⋆+∇⋆ · F⋆(U⋆, Q⋆) = 0, (8.15)
Q⋆ −∇⋆U⋆ = 0. (8.16)
This operation transforms equation (8.13) into a form that is more amenable to
treatment by the VCJH schemes.
8.3 Definitions of Viscosity
In this section, the viscosity µ which appears in the NS equations will be more pre-
cisely defined.
It is well-known that the local value of viscosity in a compressible flow of gas depends
on the type of gas and on the local value of the temperature of the flow. In fact,
the assumption that µ = const is only defensible when the Mach number is such
that the flow is in the incompressible range (e.g. when there is less than 5 percent
compressibility as is the case for M ≤ 0.3). In order to model the variations in
viscosity as a function of temperature, it is common practice to use Sutherland’s law
which takes the following form
µ = µref
(Tref + S
T + S
)(T
Tref
)3/2
, (8.17)
where µref is a reference value of the viscosity at the reference temperature Tref , and
S is ‘Sutherland’s temperature’ which is an empirically determined temperature that
depends on the type of gas.
Now, having formulated a dimensional definition for µ, it is convenient to rewrite this
definition in non-dimensional form in accordance with equation (8.11), as follows
µ⋆ =µ
µ∞=
1
Reref
(T ⋆ref + S⋆
T ⋆ + S⋆
)(T ⋆
T ⋆ref
)3/2
(8.18)
where
Reref =ρ∞u∞L
µref, T ⋆
ref =TrefT∞
, S⋆ =S
T∞. (8.19)
CHAPTER 8. GOVERNING EQUATIONS 190
This definition of the viscosity was used frequently in the numerical experiments
described in the next chapter.
Chapter 9
Results of Numerical Experiments
191
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 192
9.1 Couette Flow
In what follows, the VCJH schemes are used to solve the NS equations for the ‘Couette
flow’ problem. Couette flow occurs between two infinite walls which are unbounded
in the x − z plane, and separated by a distance of H in the y direction. One wall
is stationary with a temperature of Tw, and the other wall moves in the x direction
with a speed of Uw and (the same) temperature of Tw. For µ = const, the flow has an
analytical solution where p = const and the total energy E takes the following form
E = p
[1
γ − 1+
U2w
2R
(yH
)2
Tw + Pr U2w
2Cp
(yH
) [1−
(yH
)]]. (9.1)
It is common practice to generate approximate solutions for Couette flow on finite
domains with periodic boundary conditions imposed. In following this approach,
experiments were performed in 2D on the rectangular domain [−1, 1] × [0, 1], with
periodic conditions imposed on the left and right boundaries, and isothermal wall
conditions imposed on the upper and lower boundaries. In addition, experiments
were performed in 3D on the cuboid domain [−1, 1] × [0, 1] × [−1, 1], with periodic
conditions imposed on the left, right, front, and back boundaries, and isothermal
wall conditions imposed on the upper and lower boundaries. Figure (9.1) shows the
computational domains and their associated boundary conditions in 2D and 3D.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 193
Periodic
(left)
Periodic
(right)
Isothermal wall
(top)
Isothermal wall
(bottom)
(a) 2D rectangular domain
Periodic
(left)
Periodic
(back)
Isothermal wall
(top)
Isothermal wall
(bottom)
Periodic
(right)
Periodic
(front)
(b) 3D cuboid domain
Figure 9.1: Boundary conditions for Couette flow on 2D and 3D computational domains.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 194
For the upper and lower isothermal walls, the temperature was given the value of
T = Tw = 300K and the pressure was held constant. In addition, no-slip conditions
were imposed on the lower walls, i.e. the velocity components on the lower wall were
given the values u = 0 and v = 0 in 2D, and u = 0, v = 0, and w = 0 in 3D. Similarly,
no-slip conditions were imposed on the upper walls (each of which were required to
move at a speed of Uw in the x direction), i.e. the velocity components on the upper
wall were given the values u = Uw and v = 0 in 2D, and u = Uw, v = 0, and w = 0
in 3D.
The boundary conditions (described above) were imposed on discretizations of the
rectangular and cuboid domains in 2D and 3D. The rectangular domain in 2D was
discretized by forming 2N × N regular quadrilateral meshes and then splitting these
meshes into grids with N = 4N2 triangle elements. In this manner, structured
grids with N = 2, 4, 8, and 16 were formed. In addition, a complementary set of
unstructured triangular grids with N = 16, 64, 256, and 1024 elements was formed.
Figure (9.2) shows examples of the structured and unstructured triangular grids for
the cases of N = 4 and N = 64, respectively.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 195
x
y
-1 -0.5 0 0.5 10
0.5
1
(a) Structured grid
x
y
-1 -0.5 0 0.5 10
0.5
1
(b) Unstructured grid
Figure 9.2: Structured and unstructured triangular grids for the cases of N = 4 and N = 64.
Similarly, the cuboid domain in 3D was discretized by forming 2N × N × 2N regular
hexahedral meshes and then splitting these meshes into grids with N = 24N3 tetra-
hedron elements. In this manner, structured grids with N = 1, 2, 3, 4, 6, 8, 12, and
16 were formed. In addition, a complementary set of unstructured tetrahedral grids
with N = 24, 192, 648, 1536, 5184, 12288, 41472, and 98304 elements was formed.
Figures (9.3) and (9.4) show the structured tetrahedral grid with N = 4, as well as
the unstructured tetrahedral grid with N = 1536 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 196
X
Y
Z
(a) Structured grid
(b) Cutaway of structured grid
Figure 9.3: Structured tetrahedral grid with N = 4.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 197
X
Y
Z
(a) Unstructured grid
(b) Cutaway of unstructured grid
Figure 9.4: Unstructured tetrahedral grid with N = 1536 elements.
At time t = 0, the flow on each domain was initialized with gas parameters Pr = 0.72
and γ = 1.4, and with velocities of u = Uw and v = 0 in 2D and u = Uw, v = 0,
and w = 0 in 3D, where Uw was chosen such that the Mach number M = 0.2, and
the Reynolds number (based on H) was 200. In both 2D and 3D, the solution was
marched forward in time using the RK54 approach and, at each time-step, the inviscid
and viscous numerical fluxes were computed using the Rusanov approach (with λ = 1)
and the LDG approach (with β = ±0.5n− and τ = 0.1 in 2D and β = ±0.5n− and
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 198
τ = 1.0 in 3D). Each simulation was terminated after the residual reached machine
zero. In 2D, results were obtained on structured grids with N = 2, 4, 8, and 16 and on
unstructured grids with N = 16, 64, 256, and 1024 for polynomial orders p = 2 and 3.
In 3D, results were obtained on structured grids with N = 2, 3, 4, 6, 8, 12, and 16
for p = 2 and 3, N = 2, 3, 4, and 6 for p = 4, and N = 1, 2, and 3 for p = 5, and on
the unstructured grids with N = 192, 648, 1536, 5184, 12288, 41472, and 98304 for
p = 2 and 3, N = 192, 648, 1536, and 5184 for p = 4, and N = 24, 192, and 648 for
p = 5.
9.1.1 Orders of Accuracy and Explicit Time-Step Limits (2D)
Solutions in 2D were obtained for four different VCJH schemes defined by the fol-
lowing pairings of c and κ: (c = cdg, κ = κdg), (c = csd, κ = κsd), (c = chu, κ = κhu),
and (c = c+, κ = κ+). Tables (9.1) – (9.6) contain the absolute errors, orders of
accuracy, and maximum time-step limits for each of these schemes, and Figures (9.5)
and (9.6) show the density and energy contours obtained via the scheme with c = c+
and κ = κ+ on the unstructured mesh with N = 64 elements for the case of p = 3.
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 2 3.02e-05 5.58e-04
N = 4 3.19e-06 1.09e-04 3.24 2.36
N = 8 3.61e-07 2.37e-05 3.14 2.20
N = 16 4.37e-08 6.11e-06 3.05 1.95
csd, κsd N = 2 3.02e-05 4.94e-04
N = 4 3.19e-06 9.93e-05 3.24 2.32
N = 8 3.61e-07 2.19e-05 3.14 2.18
N = 16 4.37e-08 5.61e-06 3.05 1.96
chu, κhu N = 2 3.02e-05 4.61e-04
N = 4 3.19e-06 9.44e-05 3.24 2.29
N = 8 3.61e-07 2.10e-05 3.14 2.17
N = 16 4.37e-08 5.34e-06 3.05 1.98
c+, κ+ N = 2 3.02e-05 4.20e-04
N = 4 3.19e-06 8.90e-05 3.24 2.24
N = 8 3.61e-07 2.01e-05 3.14 2.15
N = 16 4.39e-08 5.04e-06 3.04 1.99
Table 9.1: Accuracy of VCJH schemes for the Couette flow problem on structured triangulargrids, for the case of p = 2. The inviscid and viscous numerical fluxes were computed usinga Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 199
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 2 9.54e-07 2.80e-05
N = 4 5.99e-08 3.93e-06 3.99 2.83
N = 8 3.78e-09 5.01e-07 3.99 2.97
N = 16 2.40e-10 6.11e-08 3.97 3.04
csd, κsd N = 2 9.53e-07 2.76e-05
N = 4 5.98e-08 3.88e-06 3.99 2.83
N = 8 3.77e-09 4.96e-07 3.99 2.97
N = 16 2.39e-10 6.05e-08 3.98 3.04
chu, κhu N = 2 9.53e-07 2.77e-05
N = 4 5.97e-08 3.89e-06 4.00 2.83
N = 8 3.76e-09 4.96e-07 3.99 2.97
N = 16 2.39e-10 6.05e-08 3.98 3.04
c+, κ+ N = 2 9.52e-07 2.81e-05
N = 4 5.97e-08 3.91e-06 4.00 2.84
N = 8 3.76e-09 4.99e-07 3.99 2.97
N = 16 2.38e-10 6.08e-08 3.98 3.04
Table 9.2: Accuracy of VCJH schemes for the Couette flow problem on structured triangulargrids, for the case of p = 3. The inviscid and viscous numerical fluxes were computed usinga Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
Table 9.3: Accuracy of VCJH schemes for the Couette flow problem on unstructured trian-gular grids, for the case of p = 2. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
Table 9.4: Accuracy of VCJH schemes for the Couette flow problem on unstructured trian-gular grids, for the case of p = 3. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
Table 9.5: Explicit time-step limits (∆tmax) of VCJH schemes on the structured triangulargrid with N = 8 for the Couette flow problem, for the cases of p = 2 and 3. The inviscidand viscous numerical fluxes were computed using a Rusanov flux with λ = 1 and a LDGflux with τ = 0.1 and β = ±0.5n−.
Table 9.6: Explicit time-step limits (∆tmax) of VCJH schemes on the unstructured trian-gular grid with N = 256 for the Couette flow problem, for the cases of p = 2 and 3. Theinviscid and viscous numerical fluxes were computed using a Rusanov flux with λ = 1 anda LDG flux with τ = 0.1 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 201
Figure 9.5: Contours of density obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 64 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 0.1 and β = ±0.5n−.
Figure 9.6: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 64 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 0.1 and β = ±0.5n−.
Absolute errors (and by extension orders of accuracy) were determined using L2 norms
and seminorms of the errors in the total energy E and its gradient ∇E. The time-step
limits (∆tmax) were determined using an iterative method on the structured grid with
N = 8 and on the unstructured grid with N = 256.
Tables (9.1) – (9.4) demonstrate that the expected order of accuracy is obtained for
all four schemes, for p = 2 and 3, on structured and unstructured grids. In addition,
Tables (9.5) and (9.6) demonstrate that increasing c and κ increases the maximum
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 202
time-step limit on structured and unstructured grids. In particular, for p = 2 on the
structured grid with N = 8, the scheme with c = c+ and κ = κ+ has ∆tmax = 5.56e-
03, while the collocation-based nodal DG scheme (recovered with c = 0 and κ = 0)
has ∆tmax = 3.19e-03. Similarly, for p = 2 on the unstructured grid with N = 256,
the scheme with c = c+ and κ = κ+ has ∆tmax = 4.96e-03, while the collocation-based
nodal DG scheme has ∆tmax = 2.88e-03.
9.1.2 Orders of Accuracy and Explicit Time-Step Limits (3D)
Solutions in 3D were obtained for two different VCJH schemes defined by the following
pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Tables (9.7) – (9.16)
contain the absolute errors, orders of accuracy, and maximum time-step limits for
each of these schemes, and Figures (9.7) and (9.8) show the density and energy
contours obtained via the scheme with c = c+ and κ = κ+ on the unstructured mesh
with N = 1536 elements for the case of p = 3.
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 2 3.84e-05 8.05e-04
N = 3 1.04e-05 3.06e-04 3.21 2.38
N = 4 4.18e-06 1.57e-04 3.18 2.33
N = 6 1.17e-06 6.24e-05 3.15 2.27
N = 8 4.75e-07 3.31e-05 3.12 2.20
N = 12 1.35e-07 1.41e-05 3.09 2.10
N = 16 5.62e-08 7.97e-06 3.05 1.98
c+, κ+ N = 2 3.83e-05 6.11e-04
N = 3 1.04e-05 2.45e-04 3.22 2.25
N = 4 4.16e-06 1.29e-04 3.18 2.22
N = 6 1.16e-06 5.31e-05 3.16 2.19
N = 8 4.71e-07 2.87e-05 3.13 2.14
N = 12 1.34e-07 1.23e-05 3.10 2.08
N = 16 5.55e-08 6.94e-06 3.06 1.99
Table 9.7: Accuracy of VCJH schemes for the Couette flow problem on structured tetrahe-dral grids, for the case of p = 2. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 203
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 2 1.22e-06 3.81e-05
N = 3 2.42e-07 1.19e-05 3.99 2.88
N = 4 7.68e-08 5.09e-06 3.99 2.94
N = 6 1.53e-08 1.52e-06 3.98 2.98
N = 8 4.87e-09 6.41e-07 3.98 3.01
N = 12 9.72e-10 1.88e-07 3.97 3.02
N = 16 3.10e-10 7.86e-08 3.98 3.03
c+, κ+ N = 2 1.21e-06 3.83e-05
N = 3 2.38e-07 1.19e-05 4.00 2.89
N = 4 7.53e-08 5.10e-06 4.00 2.94
N = 6 1.49e-08 1.52e-06 4.00 2.98
N = 8 4.70e-09 6.42e-07 4.00 3.00
N = 12 9.26e-10 1.89e-07 4.00 3.02
N = 16 2.93e-10 7.87e-08 4.00 3.04
Table 9.8: Accuracy of VCJH schemes for the Couette flow problem on structured tetrahe-dral grids, for the case of p = 3. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 2 2.68e-09 1.02e-07
N = 3 3.53e-10 1.91e-08 5.00 4.14
N = 4 8.39e-11 5.92e-09 4.99 4.07
N = 6 1.10e-11 1.18e-09 5.02 3.99
c+, κ+ N = 2 2.67e-09 8.96e-08
N = 3 3.51e-10 1.71e-08 5.01 4.08
N = 4 8.34e-11 5.37e-09 4.99 4.03
N = 6 1.11e-11 1.08e-09 4.97 3.95
Table 9.9: Accuracy of VCJH schemes for the Couette flow problem on structured tetrahe-dral grids, for the case of p = 4. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
c, κ Grid L2 err. L2s err. O(L2) O(L2s)
cdg, κdg N = 1 1.01e-08 2.79e-07
N = 2 1.58e-10 9.28e-09 6.01 4.91
N = 3 1.35e-11 1.19e-09 6.06 5.07
c+, κ+ N = 1 1.01e-08 2.49e-07
N = 2 1.56e-10 8.38e-09 6.01 4.89
N = 3 1.36e-11 1.11e-09 6.03 4.99
Table 9.10: Accuracy of VCJH schemes for the Couette flow problem on structured tetrahe-dral grids, for the case of p = 5. The inviscid and viscous numerical fluxes were computedusing a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.11: Accuracy of VCJH schemes for the Couette flow problem on unstructuredtetrahedral grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.12: Accuracy of VCJH schemes for the Couette flow problem on unstructuredtetrahedral grids, for the case of p = 3. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.13: Accuracy of VCJH schemes for the Couette flow problem on unstructuredtetrahedral grids, for the case of p = 4. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.14: Accuracy of VCJH schemes for the Couette flow problem on unstructuredtetrahedral grids, for the case of p = 5. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.15: Explicit time-step limits (∆tmax) of VCJH schemes for the Couette flow problemon the structured tetrahedral grids with N = 8 for the cases of p = 2 and 3, and N = 3 forthe cases of p = 4 and 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.16: Explicit time-step limits (∆tmax) of VCJH schemes for the Couette flow problemon the unstructured grids with N = 12288 for the cases of p = 2 and 3, and N = 648 forthe cases of p = 4 and 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 206
Figure 9.7: Contours of density obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 1536 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.
Figure 9.8: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the unstructured grid with N = 1536 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 207
In following the approach of the 2D experiments, absolute errors (and by extension
orders of accuracy) were determined using L2 norms and seminorms of the errors in
the total energy E and its gradient∇E. The time-step limits (∆tmax) were determined
using an iterative method on the structured grid with N = 8 for p = 2 and p = 3,
the structured grid with N = 3 for p = 4 and p = 5, the unstructured grid with
N = 12288 for p = 2 and p = 3, and the unstructured grid with N = 648 for p = 4
and p = 5.
Tables (9.7) – (9.14) demonstrate that the expected order of accuracy is obtained for
both schemes, on structured and unstructured grids for p = 2 to p = 5. In addition,
Tables (9.15) and (9.16) demonstrate that the scheme with c = c+ and κ = κ+ has
a larger time-step limit than the scheme with c = cdg and κ = κdg. In particular,
for p = 2 on the structured grid with N = 8, the scheme with c = c+ and κ = κ+
has ∆tmax = 2.73e-03, while the collocation-based nodal DG scheme has ∆tmax =
1.56e-03. Similarly, for p = 2 on the unstructured grid with N = 12288, the scheme
with c = c+ and κ = κ+ has ∆tmax = 2.53e-03, while the collocation-based nodal DG
scheme has ∆tmax = 1.48e-03.
Overall, the time-step improvements originally observed for linear problems, for larger
values of c and κ, have been shown to extend to nonlinear problems in 2D and 3D
on structured and unstructured grids. Furthermore, these improvements appear to
preserve accuracy, as the expected order of accuracy has been obtained in the vast
majority of the nonlinear experiments in this section.
9.2 Flow Generated by a Time-Dependent Source
Term
In what follows, the VCJH schemes are employed to solve the NS equations with a
time-dependent source term S. In general, the term S is incorporated into the NS
equations as follows
∂U
∂t+∇ · F(U,Q) = S, (9.2)
Q−∇U = 0. (9.3)
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 208
It turns out that for certain choices of S, the NS equations have well-known exact
solutions. In particular, Gassner et al. [103] have shown that if the source term is
defined in 2D as follows
S =
s1s2s3s4
,
s1 = (2k − ω) cos (k (x+ y)− ωt) ,s2 = cos (k (x+ y)− ωt) [(3 + 2a (γ − 1)− γ) k − ω + 2 (γ − 1) k sin (k (x+ y)− ωt)] ,s3 = cos (k (x+ y)− ωt) [(3 + 2a (γ − 1)− γ) k − ω + 2 (γ − 1) k sin (k (x+ y)− ωt)] ,
s4 =
(2γk2µ
Pr
)sin (k (x+ y)− ωt) + 2 cos (k (x+ y)− ωt)
[(1− γ + 2aγ) k − aω + (2γk − ω) sin (k (x+ y)− ωt)] , (9.4)
the following exact solution can be obtained
U =
sin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + asin (k (x+ y)− ωt) + a
(sin (k (x+ y)− ωt) + a)2
. (9.5)
Similarly, it can be shown that if the source term is defined in 3D as follows
S =
s1s2s3s4s5
,
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 209
s1 = (3k − ω) cos (k (x+ y + z)− ωt) ,
s2 =1
2cos (k (x+ y + z)− ωt)
[(9 + 4a (γ − 1)− 3γ) k − 2ω + 4 (γ − 1) k sin (k (x+ y + z)− ωt)] ,
s3 =1
2cos (k (x+ y + z)− ωt)
[(9 + 4a (γ − 1)− 3γ) k − 2ω + 4 (γ − 1) k sin (k (x+ y + z)− ωt)] ,
s4 =1
2cos (k (x+ y + z)− ωt)
[(9 + 4a (γ − 1)− 3γ) k − 2ω + 4 (γ − 1) k sin (k (x+ y + z)− ωt)] ,
s5 =
(3γk2µ
Pr
)sin (k (x+ y + z)− ωt) + 1
2cos (k (x+ y + z)− ωt)
[3 (3− 3γ + 4aγ) k − 4aω + 4 (3γk − ω) sin (k (x+ y + z)− ωt)] , (9.6)
Table 9.17: Accuracy of VCJH schemes for flow generated by a time-dependent source termon triangular grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
Table 9.18: Accuracy of VCJH schemes for flow generated by a time-dependent source termon triangular grids, for the case of p = 3. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
Table 9.19: Explicit time-step limits (∆tmax) of VCJH schemes for flow generated by atime-dependent source term on the triangular grid with N = 48, for the cases of p = 2and 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 214
Figure 9.12: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the triangular grid with N = 32 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 andβ = ±0.5n−.
Absolute errors (and by extension orders of accuracy) were determined using L2 norms
and seminorms of the errors in the total energy E and its gradient ∇E. The time-step
limits (∆tmax) were determined using an iterative method on the grid with N = 48
for p = 2 and p = 3.
Tables (9.17) and (9.18) demonstrate that the expected order of accuracy is obtained
for both schemes, for p = 2 and p = 3. In addition, Table (9.19) demonstrates that
the scheme with c = c+ and κ = κ+ has a larger time-step limit than the scheme with
c = cdg and κ = κdg. In particular, for p = 2 on the grid with N = 48, the scheme
with c = c+ and κ = κ+ has ∆tmax = 4.39e-03, while the collocation-based nodal DG
scheme has ∆tmax = 2.67e-03.
9.2.2 Orders of Accuracy and Explicit Time-Step Limits (3D)
Solutions were also obtained in 3D for the VCJH schemes with (c = cdg, κ = κdg)
and (c = c+, κ = κ+). Tables (9.20) – (9.22) contain the absolute errors, orders of
accuracy, and maximum time-step limits for each of these schemes, and Figure (9.13)
shows the energy contours obtained via the scheme with c = c+ and κ = κ+ on the
Table 9.20: Accuracy of VCJH schemes for flow generated by a time-dependent source termon tetrahedral grids, for the case of p = 2. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.21: Accuracy of VCJH schemes for flow generated by a time-dependent source termon tetrahedral grids, for the case of p = 3. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Table 9.22: Explicit time-step limits (∆tmax) of VCJH schemes for flow generated by atime-dependent source term on the tetrahedral grid with N = 48, for the cases of p = 2and 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 216
Figure 9.13: Contours of energy obtained using the VCJH scheme with c = c+ and κ = κ+on the tetrahedral grid with N = 32 for the case of p = 3. The inviscid and viscousnumerical fluxes were computed using a Rusanov flux with λ = 1 and a LDG flux withτ = 1.0 and β = ±0.5n−.
In following the approach of the 2D experiments, absolute errors (and by extension
orders of accuracy) were determined using L2 norms and seminorms of the errors in
the total energy E and its gradient∇E. The time-step limits (∆tmax) were determined
using an iterative method on the grid with N = 48 for p = 2 and p = 3.
Tables (9.20) and (9.21) demonstrate that the expected order of accuracy is obtained
for both schemes, for p = 2 and p = 3. In addition, Table (9.22) demonstrates that
the scheme with c = c+ and κ = κ+ has a larger time-step limit than the scheme with
c = cdg and κ = κdg. In particular, for p = 2 on the grid with N = 48, the scheme
with c = c+ and κ = κ+ has ∆tmax = 1.81e-03, while the collocation-based nodal DG
scheme has ∆tmax = 1.07e-03.
Overall, the time-step improvements that were observed for steady, nonlinear prob-
lems, for larger values of c and κ (cf. section 9.1), have been shown to extend to
unsteady nonlinear problems in 2D and 3D. Furthermore, these improvements appear
to preserve accuracy, as the expected order of accuracy has been obtained in the vast
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 217
majority of the nonlinear experiments in this section.
9.2.3 Aliasing Errors in 2D and 3D
Thus far, in an effort to minimize aliasing errors, all the results in this section have
been obtained with the solution and flux points placed at the locations of the quadra-
ture points from Chapter 7. Figure (9.14) illustrates the locations of the solution and
flux points on the reference triangle and tetrahedron for the case of p = 3.
(a) Point arrangement in 2D (b) Point arrangement in 3D
Figure 9.14: Placement of the solution points (circles) and flux points (squares) at thelocations of the quadrature points described in Chapter 7 for the case of p = 3.
Having established the preliminary effectiveness of the quadrature points via experi-
ments, it was important to more carefully test their ability to reduce aliasing errors.
Towards this end, the errors produced by simulations with the quadrature points
were compared to the errors produced by simulations with an alternative set of points:
namely the α-optimized points discovered by Hesthaven andWarburton [104, 105, 13].
The α-optimized points are a well-known set of points that are designed to optimize
the conditioning of the Vandermonde matrices used in discontinuous FEM procedures.
Figure (9.15) illustrates the α-optimized locations of the solution and flux points on
the reference triangle and tetrahedron for the case of p = 3.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 218
(a) Point arrangement in 2D (b) Point arrangement in 3D
Figure 9.15: Placement of the solution points (circles) and flux points (squares) at thelocations of the α-optimized points for the case of p = 3.
The α-optimized points are not designed to minimize aliasing errors, and thus, it
was instructive to compare the performance of these points to the performance of the
quadrature points. In order to evaluate the performance of the two sets of points,
each set was employed to solve equations (9.8) and (9.9) with source term parameters
Pr = 0.72, γ = 1.4, k = π, ω = π, a = 3.0, and µ = 0.001. Approximate solutions
to this problem were sought on coarsely meshed square domains with N = 12 and
N = 4 for polynomials orders p = 2 and p = 3, and coarsely meshed cube domains
with N = 12, 8, 6, and 4 for polynomial orders p = 2, 3, 4, and 5. In order to highlight
the effects of aliasing errors, the approximate solutions were computed for long times
(until time t = 10) using the scheme with c = c+ and κ = κ+. L2 errors in the energy
E for each set of points (on triangles and tetrahedra) are shown in Tables (9.23) and
(9.24), and contours of the energy obtained with the α-optimized points for the case
of N = 4 and p = 3 in 2D, and the case of N = 8 and p = 3 in 3D are shown in
Figures (9.16) and (9.17), respectively.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 219
p N L2 err. (quad-points) L2 err. (α-points) % difference
Table 9.23: Comparison of errors produced by experiments with the quadrature points andthe α-optimized points for the VCJH scheme with c = c+ and κ = κ+, for the problem withflow driven by a time-dependent source term on triangular grids, for the cases of p = 2 andp = 3. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 0.1 and β = ±0.5n−.
p N L2 err. (quad-points) L2 err. (α-points) % difference
Table 9.24: Comparison of errors produced by experiments with the quadrature points andthe α-optimized points for the VCJH scheme with c = c+ and κ = κ+, for the problem withflow driven by a time-dependent source term on tetrahedral grids, for the cases of p = 2 top = 5. The inviscid and viscous numerical fluxes were computed using a Rusanov flux withλ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
Figure 9.16: Contours of energy obtained via the VCJH scheme with c = c+, κ = κ+,and the solution and flux points placed at the locations of the α-optimized points, on thetriangular grid with N = 4 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 0.1 andβ = ±0.5n−.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 220
Figure 9.17: Contours of energy obtained via the VCJH scheme with c = c+, κ = κ+,and the solution and flux points placed at the locations of the α-optimized points, on thetetrahedral grid with N = 8 for the case of p = 3. The inviscid and viscous numericalfluxes were computed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 andβ = ±0.5n−.
Excellent examples of the aliasing errors that frequently arose in each of the simu-
lations can be seen in the form of the oscillations that distort the contours in Fig-
ures (9.16) and (9.17).
The data in Tables (9.23) and (9.24) demonstrates that the simulations with the
quadrature points produce less error than the simulations with the α-optimized points.
The reduction in error can be seen most notably for odd polynomial orders p = 3
and p = 5, where the error is reduced by roughly 25 – 60 percent. This suggests
the existence of a diffusive phenomena that tends to dampen aliasing errors for even
polynomial orders p = 2 and p = 4. More importantly, the tabulated data suggests
that (regardless of whether the order is even or odd), the quadrature points are
effective in reducing the total errors by producing smaller aliasing errors than those
produced by the α-optimized points.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 221
9.3 Flow Around Circular Cylinder
In this section, the VCJH schemes are employed to simulate the flow around a circular
cylinder. This particular flow was chosen because it is frequently used as a benchmark
for evaluating unsteady, compressible, viscous flow solvers. For low Reynolds numbers
and moderate Mach numbers, i.e., Re = 103 – 105 and M ≤ Mcritical, the freestream
flow encounters the circular cylinder, forms a boundary layer, separates, and then
forms a laminar wake of vortices commonly referred to as a ‘von Karman vortex
sheet’. The vortices in the wake are shed at a particular frequency f . The temporal
behavior of the flow is quantified by the non-dimensional ‘Strouhal’ number, which
is defined in terms of f , the cylinder diameter D, and the flow speed u as follows:
St = fD/u. In some cases, St is a known function of Re. For example, for Re = 200,
St for a circular cylinder is approximately 0.2 [106, 107, 108]. Based on this fact,
it was convenient to evaluate the VCJH schemes by simulating flow with Re = 200
around the circular cylinder.
For each VCJH scheme, the flow with Re = 200 was simulated on a 2D circular domain
with the circular cylinder located at the center. The boundaries of the domain were
placed at a distance of 100 diameters from the center of the cylinder. Isothermal wall
boundary conditions were imposed on the surface of the cylinder and characteristic
boundary conditions were imposed on the farfield boundaries. Figure (9.18) illustrates
the circular domain along with the boundary conditions that were imposed on it.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 222
Characteristic
Isothermal wall
Figure 9.18: Computational domain for the circular cylinder with associated boundaryconditions.
The domain was discretized by forming an unstructured triangular mesh with N =
63472 elements. The mesh was refined in the neighborhood of the cylinder surface in
order to resolve the boundary layer, and it was refined in the wake region in order
to resolve the von Karman vortex sheet. Figures (9.19) and (9.20) illustrate the
unstructured triangular mesh with refined boundary layer and wake regions.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 223
(a) Farfield mesh (b) Cylinder and wake mesh
Figure 9.19: Faraway views of the unstructured triangular grid with N = 63472 elementsaround the circular cylinder geometry.
(a) Cylinder mesh (b) Boundary layer mesh
Figure 9.20: Close-up views of the unstructured triangular grid with N = 63472 elementsaround the circular cylinder geometry.
The flow on each domain was initialized with gas parameters Pr = 0.72, γ = 1.4,
and Re = 200 (as mentioned previously). The flow was simulated for two different
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 224
Mach numbers: M = 0.3 and M = 0.5. After the flow velocity had been set in
accordance with the Mach number, the solution was marched forward in time using
the RK54 approach and, at each time-step, the inviscid and viscous numerical fluxes
were computed using the Rusanov approach with λ = 1 and the LDG approach
with β = ±0.5n− and τ = 0.1. Results from each simulation were evaluated after
the lift and drag reached a pseudo-periodic state. The results were obtained on the
aforementioned unstructured triangular grid with N = 63472 elements, for the case
of p = 2.
9.3.1 Time-Averaged Results
The simulations were performed with two different VCJH schemes defined by the
following pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.25)
contains the time-averaged drag coefficients and Strouhal numbers for each of these
schemes, and Figures (9.21) – (9.24) show the density, pressure, Mach number, and
vorticity contours obtained via the scheme with c = c+ and κ = κ+, for the cases of
Table 9.25: Values of the time-averaged drag coefficient and Strouhal number for the circularcylinder in flows with M = 0.3 and M = 0.5. The flows were simulated using the VCJHschemes with p = 2, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction with the Rusanovflux with λ = 1 and the LDG flux with β = ±0.5n− and τ = 0.1 on the unstructuredtriangular grid with N = 63472 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 225
(a) Density contours (b) Pressure contours
Figure 9.21: Density and pressure contours for the flow with M = 0.3 around the circularcylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 63472 elements.
(a) Mach contours (b) Vorticity contours
Figure 9.22: Mach number and vorticity contours for the flow with M = 0.3 around thecircular cylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, andp = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n−
and τ = 0.1 on the unstructured triangular grid with N = 63472 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 226
(a) Density contours (b) Pressure contours
Figure 9.23: Density and pressure contours for the flow with M = 0.5 around the circularcylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 63472 elements.
(a) Mach contours (b) Vorticity contours
Figure 9.24: Mach number and vorticity contours for the flow with M = 0.5 around thecircular cylinder. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, andp = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n−
and τ = 0.1 on the unstructured triangular grid with N = 63472 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 227
The two VCJH schemes produce values of the time-averaged drag coefficient and
Strouhal number that are within a few percent of each other. Furthermore, the
values of the Strouhal number produced by each scheme are in close agreement with
the expected value of St = 0.2. These results suggest that the VCJH schemes are
capable of accurately simulating time-dependent, strongly nonlinear, viscous flows.
In particular, these experiments have shown that the VCJH schemes are capable of
handling highly compressible (high-Mach number) flows, as Figure (9.24) shows that
the scheme with c = c+ and κ = κ+ is capable of producing stable results for a flow
in which the local Mach number exceeds M = 0.7.
9.4 Flow Around SD7003 Airfoil and Wing at 4
Degrees Angle of Attack
In what follows, the VCJH schemes are employed to simulate the flow around the
SD7003 airfoil (in 2D), and the SD7003 infinite wing (in 3D) at 4 degrees angle of
attack. Simulating the flow around these geometries is a convenient way to evaluate
the VCJH schemes, due to the abundance of data that has been collected in physical
experiments (cf. [109] and [110]) and numerical experiments (cf. [111], [112], [113],
[114], and [115]) on these geometries. In 2D, the SD7003 geometry is that of a low
Reynolds number Selig/Donovan (SD) airfoil that has a maximum thickness of 9.2
percent at 30.9 percent chord, and a maximum camber of 1.2 percent at 38.3 percent
chord [116]. Figure (9.25) shows an illustration of the SD7003 airfoil with a rounded
trailing edge, which is a version of the airfoil that is frequently used in numerical
experiments.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 228
0 0.2 0.4 0.6 0.8 1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(a) Airfoil geometry
0.975 0.98 0.985 0.99 0.995
−0.01
−0.005
0
0.005
0.01
(b) Trailing edge
Figure 9.25: SD7003 airfoil section (left) and a magnified view of its rounded trailing edge(right).
A SD7003 wing can be formed by extruding the SD7003 airfoil geometry. In fact, it is
common-practice to approximate the infinite SD7003 wing by extruding the SD7003
airfoil a finite distance (0.2c where c is the chord) in the z direction and prescribing
periodic boundary conditions on the x − y planes that border the resulting wing-
section. Figure (9.26) presents an illustration of the finite SD7003 wing-section with
a span of 0.2c.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 229
(a) Wing geometry (b) Trailing edge
Figure 9.26: SD7003 wing-section (left) and a magnified view of its rounded trailing edge(right).
Numerical experiments on the SD7003 geometry in 2D were performed on a circular
domain with a radius of 50c, centered at the leading edge of the airfoil. Characteristic
boundary conditions were prescribed on the outermost edge of the circular domain
and adiabatic wall boundary conditions were prescribed on the surface of the airfoil.
Similarly, numerical experiments on the SD7003 geometry in 3D were performed on
a cylindrical domain formed by extruding a circular domain (with a radius of 20c) by
0.2c in the z direction. Periodic boundary conditions were prescribed on the front and
back of the cylindrical domain, characteristic boundary conditions were prescribed on
the sides of the domain, and adiabatic wall boundary conditions were prescribed on
the surface of the wing-section. Figure (9.27) illustrates the boundary conditions that
were prescribed on the computational domains in 2D and 3D.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 230
Adiabatic wall
Characteristic
(a) 2D circular domain
Periodic (front)
Periodic (back)
Adiabatic wall
Characteristic (side)
(b) 3D cylindrical domain
Figure 9.27: Boundary conditions for 2D and 3D computational domains for simulating theSD7003 airfoil and wing-section. Note: The domains are not drawn to scale.
The 2D computational domain was discretized into an unstructured grid with N =
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 231
25810 triangular elements, and the 3D computational domain was discretized into
an unstructured grid with N = 711332 tetrahedral elements. Figures (9.28) – (9.31)
illustrate the unstructured triangular and tetrahedral grids.
(a) Farfield mesh (b) Airfoil and wake mesh
Figure 9.28: Faraway views of the unstructured triangular grid with N = 25810 elementsaround the SD7003 airfoil geometry.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 232
(a) Airfoil mesh (b) Boundary layer mesh
Figure 9.29: Close-up views of the unstructured triangular grid with N = 25810 elementsaround the SD7003 airfoil geometry.
(a) Farfield mesh (b) Wing and wake mesh
Figure 9.30: Faraway views of the unstructured tetrahedral grid with N = 711332 elementsaround the SD7003 wing geometry.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 233
(a) Wing mesh (b) Boundary layer mesh
Figure 9.31: Close-up views of the unstructured tetrahedral grid with N = 711332 elementsaround the SD7003 wing geometry.
It should be noted, that, in order to facilitate the accurate representation of the
SD7003 geometries, the edges and faces of the triangular and tetrahedral elements on
each grid were defined in terms of 2nd-order, quadratic polynomials in 2D and 3D,
respectively.
At time t = 0, a uniform flow with the properties of air (Pr = 0.72, γ = 1.4) was
initialized on each unstructured grid. The incoming flow was given a Mach number of
M = 0.2, and an angle of entry of 4 (in order to simulate flight at an angle of attack
of α = 4). In 2D, the flow was initialized with Reynolds numbers (based on c) of
Re = 10000, Re = 22000, and Re = 60000, and in 3D the flow was initialized with
Re = 10000. In both 2D and 3D, the solution was marched forward in time using
the RK54 approach and, at each time-step, the inviscid and viscous numerical fluxes
were computed using the Rusanov approach (with λ = 1) and the LDG approach
(with β = ±0.5n− and τ = 0.1 in 2D and β = ±0.5n− and τ = 1.0 in 3D). Results
from each simulation were evaluated after the lift and drag reached a pseudo-periodic
state. In 2D, results were obtained on the unstructured grid with N = 25810 for
p = 2, and in 3D, results were obtained on the unstructured grid with N = 711332
for p = 3.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 234
9.4.1 Lift and Drag Results (2D)
Solutions were obtained in 2D using the VCJH schemes defined by the following
pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.26) compares
the time-averaged values of the lift and drag coefficients for each of the schemes with
the values from [115]. In addition, Figures (9.32) – (9.34) show portions of the time
histories of the lift and drag coefficients obtained via the scheme with c = c+ and
κ = κ+, and Figures (9.35) – (9.37) show the density and vorticity contours obtained
Table 9.26: Time-averaged values of the lift and drag coefficients for the SD7003 airfoil inflows with Re = 10000, Re = 22000, and Re = 60000. The flows were simulated usingthe VCJH schemes with p = 2, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction withthe Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− and τ = 0.1 on theunstructured triangular grid with N = 25810 elements.
75 76 77 78 79 80
0.34
0.36
0.38
0.4
0.42
t
CL
(a) Lift coefficient
75 76 77 78 79 80
0.046
0.048
0.05
0.052
0.054
t
CD
(b) Drag coefficient
Figure 9.32: Temporal variation of the lift and drag coefficients for the SD7003 airfoil inflow with Re = 10000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 0.1 on the unstructured triangular grid with N = 25810 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 235
70 75 800.3
0.4
0.5
0.6
0.7
0.8
0.9
t
CL
(a) Lift coefficient
70 75 800.01
0.02
0.03
0.04
0.05
0.06
0.07
t
CD
(b) Drag coefficient
Figure 9.33: Temporal variation of the lift and drag coefficients for the SD7003 airfoil inflow with Re = 22000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 0.1 on the unstructured triangular grid with N = 25810 elements.
70 75 800.45
0.5
0.55
0.6
0.65
0.7
t
CL
(a) Lift coefficient
70 75 800.01
0.015
0.02
0.025
0.03
0.035
t
CD
(b) Drag coefficient
Figure 9.34: Temporal variation of the lift and drag coefficients for the SD7003 airfoil inflow with Re = 60000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 0.1 on the unstructured triangular grid with N = 25810 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 236
(a) Density contours (b) Vorticity contours
Figure 9.35: Density and vorticity contours for the flow with Re = 10000 around the SD7003airfoil. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 25810 elements.
(a) Density contours (b) Vorticity contours
Figure 9.36: Density and vorticity contours for the flow with Re = 22000 around the SD7003airfoil. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 25810 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 237
(a) Density contours (b) Vorticity contours
Figure 9.37: Density and vorticity contours for the flow with Re = 60000 around the SD7003airfoil. The flow was simulated using the VCJH scheme with c = c+, κ = κ+, and p = 2in conjunction with the Rusanov flux with λ = 1 and the LDG flux with β = ±0.5n− andτ = 0.1 on the unstructured triangular grid with N = 25810 elements.
The data in Table (9.26) demonstrates that the results produced by the VCJH schemes
are in close agreement with the results independently obtained by [115]. Overall,
based on the tabulated results and the results in Figures (9.32) – (9.37), it appears
that the VCJH schemes are successful in simulating changes in the frequencies and
amplitudes of the lift and drag forces as a function of Reynolds number, and in
accurately representing the rapidly evolving vortex structures that are formed in the
shear layer that emanates from the trailing edge of the airfoil.
9.4.2 Lift and Drag Results (3D)
Solutions were obtained in 3D using the VCJH schemes defined by the following
pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Table (9.27) compares
the time-averaged values of the lift and drag coefficients for these schemes with the
values from [115]. In addition, Figure (9.38) shows portions of the time histories of
the lift and drag coefficients obtained via the scheme with c = c+ and κ = κ+, and
Figure (9.39) shows isosurfaces of the density and vorticity obtained via the scheme
Table 9.27: Time-averaged values of the lift and drag coefficients for the SD7003 wing-section in a flow with Re = 10000. The flow was simulated using the VCJH schemes withp = 3, c = cdg, κ = κdg and c = c+, κ = κ+ in conjunction with the Rusanov flux withλ = 1 and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedralgrid with N = 711332 elements.
40 41 42 43 44 450.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
t
CL
(a) Lift coefficient
40 41 42 43 44 450.046
0.047
0.048
0.049
0.05
0.051
0.052
0.053
t
CD
(b) Drag coefficient
Figure 9.38: Temporal variation of the lift and drag coefficients for the SD7003 wing-sectionin flow with Re = 10000. The flow was simulated using the VCJH scheme with c = c+,κ = κ+, and p = 3 in conjunction with the Rusanov flux with λ = 1 and the LDG flux withβ = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid with N = 711332 elements.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 239
(a) Density isosurfaces (b) Vorticity isosurfaces
Figure 9.39: Density and vorticity isosurfaces colored by Mach number for the flow withRe = 10000 around the SD7003 wing-section. The flow was simulated using the VCJHscheme with c = c+, κ = κ+, and p = 3 in conjunction with the Rusanov flux with λ = 1and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid withN = 711332 elements.
The data in Table (9.27) demonstrates that the results produced by the VCJH schemes
are in close agreement with the results obtained by [115]. Thus, the favorable perfor-
mance that was originally observed in 2D on the SD7003 airfoil appears to extend to
3D on the SD7003 wing-section.
9.5 Flow Around SD7003 Wing at 30 Degrees An-
gle of Attack
In this section, the VCJH schemes are employed to simulate flow around an infinite
SD7003 wing at 30 degrees angle of attack. For this case, the incoming freestream
encounters the wing and forms a large wake of unsteady, separated, vortical flow.
In [114], Persson successfully simulated the flow using a DG scheme in conjunction
with a hybrid implicit-explicit time-stepping scheme. In this thesis, the focus is on
successfully employing the VCJH schemes in conjunction with explicit time-stepping
schemes, and thus (naturally) an explicit time-stepping scheme was utilized during
each simulation of the flow with the VCJH schemes. The objective of the simulations
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 240
was not to compare the computational cost of the VCJH schemes in conjunction
with an explicit time-stepping scheme to the computational cost of Persson’s DG
scheme in conjunction with a hybrid implicit-explicit time-stepping scheme. Rather,
the objective was to demonstrate that the VCJH schemes in conjunction with an
explicit time-stepping scheme could produce stable results for a highly-separated,
vortex-dominated flow.
In the numerical experiments, the flow was simulated on a cuboid domain Ω =
on the front and back faces of the domain, characteristic boundary conditions were
imposed on the left, right, top and bottom faces of the domain, and adiabatic wall
boundary conditions were imposed on the wing-section. Figure (9.40) shows the
domain Ω along with the associated boundary conditions.
Adiabatic wall
Characteristic(right)
Characteristic (bottom)
Characteristic(left)
Periodic (back)
Periodic (front)
Characteristic (top)
Figure 9.40: Boundary conditions for the computational domain for simulating the SD7003wing-section. Note: The domain is not drawn to scale.
The domain was discretized into an unstructured grid of N = 311958 tetrahedral
elements. The grid, which was similar to the one that was utilized in [114], was
generously provided by Persson. Figure (9.41) illustrates the grid.
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 241
(a) Farfield mesh
(b) Airfoil and wake mesh (c) Boundary layer mesh
Figure 9.41: Views of the unstructured tetrahedral grid with N = 311958 elements aroundthe SD7003 wing-section.
It should be noted, that, in order to facilitate the accurate representation of the
SD7003 geometry, the faces of the tetrahedral elements on the grid were defined in
terms of 2nd-order, quadratic polynomials.
At time t = 0, a uniform flow with the properties of air (Pr = 0.72, γ = 1.4)
was initialized on the unstructured tetrahedral grid. The incoming flow was given
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 242
a Mach number of M = 0.2 and a Reynolds number of Re = 100000. The solution
was marched forward in time using the RK54 approach and, at each time-step, the
inviscid and viscous numerical fluxes were computed using the Rusanov approach
with λ = 1 and the LDG approach with β = ±0.5n− and τ = 1.0. Results were
obtained on the aforementioned unstructured grid with N = 311958 for polynomial
order p = 2 at time t = 3.
9.5.1 Isosurfaces and Time-Step Limits
Solutions were obtained using the VCJH schemes defined by the following pairings
of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Figure (9.42) shows the density
and vorticity isosurfaces obtained via the scheme with c = c+ and κ = κ+.
(a) Density isosurfaces (b) Vorticity isosurfaces
Figure 9.42: Density and vorticity isosurfaces colored by Mach number for the flow withRe = 100000 around the SD7003 wing-section. The flow was simulated using the VCJHscheme with c = c+, κ = κ+, and p = 2 in conjunction with the Rusanov flux with λ = 1and the LDG flux with β = ±0.5n− and τ = 1.0 on the unstructured tetrahedral grid withN = 311958 elements.
Figure (9.42) demonstrates that the VCJH scheme with c = c+ and κ = κ+ is capable
of producing a stable simulation of the initial vortices that are generated by the wing
at time t = 3. In addition, further experiments demonstrated that the VCJH scheme
with c = c+ and κ = κ+ has a larger time-step limit than the scheme with c = cdg
CHAPTER 9. RESULTS OF NUMERICAL EXPERIMENTS 243
and κ = κdg. In particular, an iterative method was used to show that the VCJH
scheme with c = c+ and κ = κ+ has ∆tmax = 5.09e-06, while the collocation-based
nodal DG scheme has ∆tmax = 2.84e-06.
Chapter 10
Conclusion
244
CHAPTER 10. CONCLUSION 245
10.1 Summary
A new class of energy stable FR schemes has been proposed for triangular and tetra-
hedral elements. These schemes (referred to as VCJH schemes) utilize the VCJH
correction functions and fields in order to correct the solution and the flux in the
advection-diffusion equation. It has been shown that these schemes are provably
stable for linear advection-diffusion problems in 2D and 3D, for all orders of accu-
racy on unstructured grids. In addition, it has been shown that the schemes can be
parameterized by two constant scalars c and κ, and that for appropriate choices of
these scalars, a class of filtered collocation-based nodal DG schemes can be recovered.
Furthermore, numerical experiments have demonstrated that the VCJH schemes are
stable and accurate for linear model problems, and that a certain VCJH scheme pos-
sesses explicit time-step limits that are (in some cases) more than 2x greater than
those of the collocation-based nodal DG scheme.
The behavior of the VCJH schemes was also investigated in the context of nonlin-
ear advection-diffusion problems. Here, it was shown that the VCJH schemes are
guaranteed to be stable if the nonlinearity of the flux is sufficiently weak, and if the
approximate flux is constructed using an exact L2-projection that eliminates all alias-
ing errors. In addition, if the approximate flux is constructed using a (less expensive)
collocation-projection instead of an L2-projection, it was shown that placing the so-
lution and flux points at the locations of quadrature points minimizes aliasing errors.
As a result, it was recommended that the solution and flux points be placed at the
locations of a class of newly discovered quadrature points on triangles and tetrahedra.
Finally, a series of numerical experiments on nonlinear problems were executed in
order to assess the practical performance of the VCJH schemes in conjunction with the
new quadrature points. The behavior of the schemes was evaluated on structured and
unstructured grids around straight-edged and curved-edged geometries. Overall, the
results of the experiments indicated that the favorable properties of the VCJH schemes
that were initially observed for linear problems (i.e. the stability and increased time-
step limits that were initially observed for linear problems) extended to nonlinear
problems. Furthermore, it was shown that the schemes are capable of obtaining high-
order accuracy and producing results that are in good agreement with independently
performed numerical experiments.
CHAPTER 10. CONCLUSION 246
10.2 Future Avenues for Research
A natural extension of this research entails adapting the VCJH schemes to simulate a
wider range of fluid-flow phenomena. In particular, the current schemes are tailored
towards obtaining high-order approximate solutions to unsteady linear and nonlin-
ear problems that have relatively smooth exact solutions (i.e unsteady linear and
nonlinear problems whose exact solutions are differentiable at least once). Further
efforts must be employed to extend the schemes to treat problems involving flows
with sharp discontinuities, such as flows with shock waves. Furthermore, additional
efforts are required in order to extend the VCJH schemes to mixed grids of triangular
and quadrilateral elements in 2D and hexahedral and tetrahedral elements in 3D. In
particular, tensor product extensions of the 1D VCJH schemes to quadrilateral and
hexahedral grids have yet to be proven stable. In addition, the VCJH schemes have
yet to be proven stable for transitional elements that are frequently required for the
construction of mixed grids, such as prisms and pyramids.
Although the VCJH schemes have been tailored towards treating unsteady problems,
there may be some benefit in a detailed evaluation of the schemes’ behavior for steady
problems. In the midst of unsteady problems, there are frequently regions in which
the flow is predominantly steady. Thus far, the stability of the schemes for quasi-
steady, or steady flows has yet to be investigated in rigorous mathematical detail. In
particular, the VCJH schemes have yet to be proven stable for the canonical steady-
state advection-diffusion problem, i.e. the linear advection-diffusion problem with a
Table C.1: Quadrature rules with Np = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points forthe triangle (the 2-simplex). Note that the quadrature point locations for these rules areconsistent with the 2-SCP configurations.
Table C.2: Quadrature rules with Np = 45, 55, and 66 points for the triangle (the 2-simplex). Note that the quadrature point locations for these rules are inconsistent with the2-SCP configurations.
Table C.3: Quadrature rule with Np = 84 points for the tetrahedron (the 3-simplex).
APPENDIX 265
Appendix D. Experiments on a Steady Linear
Advection-Diffusion Problem
In this section, the VCJH schemes are employed to solve the steady linear advection-
diffusion equation in 3D which takes the following form
∇ · (au− bq) = S, (D.1)
q−∇u = 0. (D.2)
Here, it should be noted that the advection and diffusion of the scalar u is driven
by the source term S. There is a steady solution to equations (D.1) and (D.2) if the
boundary conditions on u are well-posed, and if the source term S is constant in time
and (at most) varies in space.
D.1 Problem Definition
In particular, a solution to equations (D.1) and (D.2) can be obtained within an
L-shaped domain Ω = [0, 1]× [0, 1]×[−1
2, 12
]\[−1
2, 12
]×[12, 1]×[−1
2, 12
](shown in
Figure (D.1)), if no-slip Dirichlet boundary conditions are imposed on all faces of the
domain, i.e. u = 0 is enforced on all of Γ, and flow within the domain is driven by
the source term S = 1.
APPENDIX 266
x
y
z
Figure D.1: L-shaped 3D computational domain.
The resulting steady flow does not have an analytical solution, however, it does yield
numerical solutions which have been studied in considerable detail by Brogniez [119].
It turns out that the resulting steady flow is a useful test of a scheme’s stability.
In particular, for a = (cos θ, sin θ, 0), θ = π/6, and b = 5 × 10−3, the flow has
an extremely thin boundary-layer region at x = 1. This region frequently causes
overshoots or undershoots to appear and these spurious oscillations may cause the
approximate solution to diverge during the course of its iterative march towards the
final steady state. As a result, it was deemed necessary to perform experiments on
this problem with the VCJH schemes in order to evaluate the schemes’ robustness
in the presence of this particular phenomena. For the experiments, the domain Ω
was tessellated with an unstructured tetrahedral mesh with N = 144847 elements as
shown in Figure (D.2).
APPENDIX 267
(a) Transparent view (b) Solid view
Figure D.2: Unstructured tetrahedral mesh with N = 144847 elements.
At time t = 0, the flow on the domain was initialized with velocity magnitude
|a| = 1. The solution was marched forward in time towards a steady state using
the RK54 approach and, at each time-step, the inviscid and viscous numerical fluxes
were computed using the Rusanov approach with λ = 1 and the LDG approach with
β = ±0.5n− and τ = 1.0. Each simulation was terminated after the residual reached
machine zero. Steady solutions were obtained on the aforementioned unstructured
tetrahedral grid for polynomials orders p = 2 to p = 5.
D.2 Steady-State Results
Solutions were obtained for two different VCJH schemes defined by the following
pairings of c and κ: (c = cdg, κ = κdg) and (c = c+, κ = κ+). Figures (D.3) and
(D.4) show the velocity profiles for the schemes, and Figure (D.5) shows the solution
contours obtained via the scheme with c = c+ and κ = κ+.
APPENDIX 268
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
c = c
dg, κ = κ
dg
c = c+, κ = κ
+
(a) p = 2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
(b) p = 3
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
(c) p = 4
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
(d) p = 5
Figure D.3: Velocity profiles at y = 0.3 and z = 0 for the steady linear advection-diffusionproblem, for the cases of p = 2 to p = 5. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
APPENDIX 269
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
c = c
dg, κ = κ
dg
c = c+, κ = κ
+
(a) p = 2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
(b) p = 3
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
(c) p = 4
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
(d) p = 5
Figure D.4: Velocity profiles at x = 0.9 and z = 0 for the steady linear advection-diffusionproblem, for the cases of p = 2 to p = 5. The inviscid and viscous numerical fluxes werecomputed using a Rusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
APPENDIX 270
(a) p = 2 (b) p = 3
(c) p = 4 (d) p = 5
Figure D.5: Solution contours for the steady linear advection-diffusion problem, for thecases of p = 2 to p = 5. The inviscid and viscous numerical fluxes were computed using aRusanov flux with λ = 1 and a LDG flux with τ = 1.0 and β = ±0.5n−.
For smaller values of the polynomial order p, i.e., p = 2 and p = 3, the schemes
produce weak spurious oscillations in the velocity profiles. However, for p = 4 and
p = 5, the velocity profiles are devoid of oscillations, indicating that the boundary
layer at x = 1 is now sufficiently resolved. Most importantly, for all cases, for even
the smaller values of p, both VCJH schemes remain stable. Overall, the favorable
properties of the schemes with regard to stability, that were originally observed for
unsteady linear problems, have been shown (in this case) to extend to steady linear
problems.
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