Journal of Computational Physicsmakxu/PAPER/GKS-DG.pdf178 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 Since the mass, momentum, and energy are conserved during

Journal of Computational Physics 292 (2015) 176–193

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Journal of Computational Physics

www.elsevier.com/locate/jcp

A multi-dimensional high-order discontinuous Galerkinmethod based on gas kinetic theory for viscous flow

computations

Xiaodong Ren a, Kun Xu a,b,∗, Wei Shyy b, Chunwei Gu c

a Department of Mathematics, School of Science, Hong Kong University of Science and Technology, Hong Kong, Chinab Department of Mechanical and Aerospace Engineering, School of Engineering, Hong Kong University of Science and Technology, Hong Kong, Chinac Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 September 2014Received in revised form 13 March 2015Accepted 17 March 2015Available online 24 March 2015

Keywords:Discontinuous GalerkinGas-kinetic schemeNavier–Stokes equationsLinear Least Square

This paper presents a high-order discontinuous Galerkin (DG) method based on a multi-dimensional gas kinetic evolution model for viscous flow computations. Generally, the DG methods for equations with higher order derivatives must transform the equations into a first order system in order to avoid the so-called “non-conforming problem”. In the traditional DG framework, the inviscid and viscous fluxes are numerically treated differently. Differently from the traditional DG approaches, the current method adopts a kinetic evolution model for both inviscid and viscous flux evaluations uniformly. By using a multi-dimensional gas kinetic formulation, we can obtain a spatial and temporal dependent gas distribution function for the flux integration inside the cell and at the cell interface, which is distinguishable from the Gaussian Quadrature point flux evaluation in the traditional DG method. Besides the initial higher order non-equilibrium states inside each control volume, a Linear Least Square (LLS) method is used for the reconstruction of smooth distributions of macroscopic flow variables around each cell interface in order to construct the corresponding equilibrium state. Instead of separating the space and time integrations and using the multistage Runge–Kutta time stepping method for time accuracy, the current method integrates the flux function in space and time analytically, which subsequently saves the computational time. Many test cases in two and three dimensions, which include high Mach number compressible viscous and heat conducting flows and the low speed high Reynolds number laminar flows, are presented to demonstrate the performance of the current scheme.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

In order to improve the reliability of numerical methods and present accurate flow computations, the development of high order (>2nd) schemes has been under intensive investigation recently. Most finite volume (FV) methods are based on the piecewise constant representation of flow variables and resort the reconstruction techniques to obtain high order

* Corresponding author at: Department of Mathematics, School of Science, Hong Kong University of Science and Technology, Hong Kong, China.E-mail addresses: [email protected] (X. Ren), [email protected] (K. Xu), [email protected] (W. Shyy), [email protected] (C. Gu).

http://dx.doi.org/10.1016/j.jcp.2015.03.0310021-9991/© 2015 Elsevier Inc. All rights reserved.

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 177

accuracy in space. Generally these methods are effective for structured meshes, but may face reconstruction problem on arbitrary grids in the multidimensional unstructured mesh cases due to the use of extended stencils. In order to avoid difficulties in the reconstruction, one possible way is to develop compact methods, and the discontinuous Galerkin (DG) method [1] becomes one of these idealized choices. For the DG method, the higher accuracy is achieved by means of higher order polynomial approximation inside each cell. Only the information from adjacent cells with common cell interfaces isneeded for the update of the degree of freedom of the cell. Therefore, the DG method can deliver higher order accurate solutions without solely relying on the reconstruction techniques and large stencils.

The DG method was firstly introduced by Reed and Hill [2] and applied to a linear transport equation by Lesaint and Raviart [3]. Chavent and Salzano [4] firstly adapted the method to a nonlinear case. Cockburn and Shu [1,5] further devel-oped the method in a series of papers, in which a framework to solve the nonlinear time dependent hyperbolic conservation laws was established. For convection–diffusion equations, the DG method proposed by Cockburn and Shu [1,5] cannot be directly applied, because the discontinuities appearing at the cell interface are not regular enough to handle higher order derivatives [6]. The alternative formulation proposed by Bassi and Rebay [7] is to first transform the convection–diffusion equations into a first order system. This technique successfully extends the DG method to the Navier–Stokes (NS) flow computation. Cockburn and Shu [8] proposed a local discontinuous Galerkin method based on a similar formulation.

Alternatively, the Navier–Stokes solutions can be recovered using a gas-kinetic formulation [9–13], where a kinetic flux function including both inviscid and viscous terms can be obtained in the kinetic evolution model. In the gas kinetic scheme, the fluxes are constructed based on the integral solution of the gas kinetic Bhatnagar–Gross–Krook (BGK) model, which presents a multiscale evolution process from a non-equilibrium to an equilibrium state, with the inclusion of time evolution of both flow variables and their derivatives. Different from the Riemann problem, the flow dynamics from a higher order initial reconstruction is explicitly followed. Xu [12] firstly proposed a one dimensional DG method by using a 2nd order BGK scheme for the flux computation. Liu and Xu [9] adopted the 2nd order BGK scheme on each Gaussian Quadrature point, where the directional splitting method is used for the 2D cases. Ni et al. [11] and Luo et al. [10] also used the 2nd order BGK scheme for the flux evaluation. In this paper, we will use a multi-dimensional 3rd order gas-kinetic BGK scheme, which is similar with the finite volume BGK scheme [14], to evaluate the fluxes both inside a cell and at a cell interface. Based on the multi-dimensional gas kinetic formulation, a spatial and temporal dependent gas distribution function can be evaluated explicitly, which can be integrated analytically in both space and time without using the integration formula based on the Gaussian Quadrature points. In the current approach, an LLS method is used for the equilibrium state construction around a cell interface, which is approached by the initial non-equilibrium state in the time evolution process.

The paper is organized as follows. Section 2 is the construction of the numerical method, which is composed of the gas kinetic discontinuous Galerkin scheme, the gas distribution function construction, and an LLS method for the equilibrium state construction. Section 3 shows some numerical examples and the results. The last section draws the conclusion.

2. Numerical method

2.1. Gas kinetic discontinuous Galerkin scheme

A 3D gas-kinetic BGK model is∂ f

∂t+ �u • ∇ f = g − f

τ, (1)

where �u = (u, v, w) is the particle velocity vector, f is the gas distribution function, g is the equilibrium state approached by f , ∇ f is the gradient of f with respect to x = (x, y, z), τ is the particle collision time which is related to the viscosity and heat conduction coefficients, and t is the physical time.

The equilibrium state is a Maxwellian distribution,

g = ρ

(λ

π

) K+32

e−λ[(u−U )2+(v−V )2+(w−W )2+ξ2], (2)

in which ρ is the density, U , V and W are the macroscopic velocity in x-, y- and z-direction, and λ is related to the gas temperature T by λ = m/2kT , where m is the molecular mass and k is the Boltzmann constant. The total number of degree of freedom K in the internal variable ξ is equal to (5 − 3γ )/(γ − 1) and γ is the specific heat ratio. In the above equilibrium state g , the internal variable ξ2 is equal to ξ2 = ξ2

1 + ξ22 + · · · + ξ2

K . The relation between the macroscopic conservative variables and the distribution function is

Q = (ρ,ρU ,ρV ,ρW ,ρE)T =∫

ψ f dΞ =∫

ψ gdΞ, (3)

where the vector of moments ψ is

ψ =(

1, u, v, w,1

2

(u2 + v2 + w2 + ξ2))T

, (4)

and dΞ = dudvdwdξ1dξ2 · · ·dξK is the volume element in the phase space.

178 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Since the mass, momentum, and energy are conserved during particle collisions, f and g satisfy the conservation con-straint at any point in space and time,∫

ψg − f

τdΞ = 0, (5)

where τ is assumed to be independent of the particle velocity.Based on the above BGK model and the Chapman–Enskog expansion, the corresponding NS equations can be derived

[13]. The Chapman–Enskog expansion for the NS distribution function is

fNS(x, t, u, v, w, ξ) = g − τ

(∂ g

∂t+ u

∂ g

∂x+ v

∂ g

∂ y+ w

∂ g

∂z

), (6)

and the relation between the flux and the distribution function is

�F =∫

�uψ f dΞ. (7)

Integrating Eq. (1) to the moments ψ and using Eqs. (3) and (7), we have

∂ Q (x, t)

∂t+ ∇ • �F = 0, x ∈ Ω, (8)

where Ω indicates the flow domain. Note that Eq. (8) describes a general conservation law, which does not imply the equilibrium limit (τ → 0), i.e. f = g for the Euler equations. The zero on the right side is due to the conservation property in Eq. (5), which is valid for the update of conservative variables.

In order to obtain the numerical approximate solution of Q , the domain Ω is decomposed into a finite number of cells and expressed as

Ω.=

m⋃e=1

Ωe = {Ω1,Ω2, · · · ,Ωm}, m ∈N, (9)

where m is the cell number.A local polynomial basis {υ0 = 1, υ ′

1, · · · , υ ′l−1} is adopted to define the approximate polynomial solution in each cell

Q.= Q h(x, t) = Q (t) +

l−1∑i=1

Q i(t)(υ ′

i (x) − υ ′i

), x ∈ Ωe, t ≥ 0, l ∈N, (10)

in which l is the number of basis functions, Q (t) is the averaged conservative variables, and υ ′i is the mean value of the

basis function within the cell, which has the following definition

υ ′i = 1

|Ωe|∫Ωe

υ ′i dx, (11)

where |Ωe| is the cell volume.Rewrite the basis functions as υi = υ ′

i − υ ′i and Eq. (10) can be rewritten as

Q h(x, t) = Q (t) +l−1∑i=1

Q i(t)υi(x). (12)

Generally speaking, different kinds of basis functions can be used. Considering the generality of cell type, we use a Taylor series expansion at the barycenter of the cell to represent the approximate solutions in this paper [15].

With the mean values of the conservative variables at the nth time step, a finite volume method can be used to update the cell averaged values at the next time step:

( Q )n+1 = ( Q )n − 1

|Ωe|∫

∂Ωe

( �t∫0

�F dt

)• �ndS, (13)

where ∂Ωe is the cell interface, �n is the outward normal unit vector of the corresponding interface, and �t is the time step.To evaluate the high order parts of the approximate solutions, we use the DG method. Substituting Eq. (12) into Eq. (8)

and using the Galerkin method, we have∫∂( Q (t) +∑l−1

i=1 Q i(t)υi(x))

∂tυ j(x)dx +

∫∇ • �Fυ j(x)dx = 0, j = 1, · · · , l − 1. (14)

Ωe Ωe

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 179

Using the definition in Eq. (11) and the integration by parts, we obtain

l−1∑i=1

υiυ j(

Qn+1i − Q

ni

)= 1

|Ωe|∫Ωe

( �t∫0

�F dt

)• ∇υ j(x)dx − 1

|Ωe|∫

∂Ωe

( �t∫0

�F dt

)• �nυ j(x)dS, j = 1, · · · , l − 1, (15)

where υiυ j = 1|Ωe |

∫Ωe

υi(x)υ j(x)dx.For problems with the discontinuity, the above method would cause nonlinear instability around the discontinuous

region. Here we adopt a shock detector and a compact limiter to prevent the oscillatory instability. The shock detector is similar to the one proposed in [16]. It depends on the cell interface values (Q + and Q −) at both sides of the interface. Taking the cell Ωe as an example, the shock detector parameter is defined as

ISD = I ′SD

λASD, (16)

I ′SD =∑

i, �V •�n<0

1

(hi)q

∫∂Ωe,i

Q ′dS, ASD =∑

i, �V •�n<0

|∂Ωe,i|, (17)

Q ′ = ∣∣2(Q +i − Q −

i

)/(Q +

i + Q −i

)∣∣, (18)

where i is the index of the interface of cell Ωe , ∂Ωe,i is the i-th interface of the cell, and |∂Ωe,i | is its area. Here �V is the characteristic flow velocity vector at the interface, hi is the local width of the cell Ωe , q = (p + 1)/2, and λ = 5p−1, where p is the order of the approximate polynomial. The parameter Q is chosen as Q = ρE . By using the definition, the detection scheme is

ϕ ={

1.0, ISD > 1 or ρminΩe

≤ 0 or pminΩe

≤ 0, “troubled cell”

0.0, else, “untroubled cell”.(19)

Once the troubled cells are identified, a compact limiter is used for the polynomial solutions correction in these troubled cells. The objective of the limiter is to determine a suitable parameter α j to limit the j-th conservative flow variable through the following constraint,

Q minj ≤ Q j = Q j + α j • δQ j(x) ≤ Q max

j , (20)

where Q j denotes the mean value and δQ j is the variation.In order to determine the parameter vector α, we should firstly estimate Q min and Q max. There are several methods can

be used, such as methods in [17–19]. Here, we adopt a compact strategy to obtain the two vectors and they are described in the following.

For any troubled cell Ωe , we denote the solution polynomials in this cell and its adjacent cells sharing a common interface as

Q 0 = Q 0 + δ Q 0, Q i = Q i + δ Q i (i = 1, · · ·), (21)

where the subscript “0” means the troubled cell and “i” means the i-th adjacent cell sharing the i-th interface with the troubled cell. In order to maintain the original cell mean values, we make the following modification:

Q i = Q i − Q i + Q 0 = Q 0 + Q i − Q i + δ Q i = Q 0 + δ Q i, (22)

where Q i is the mean value of Q i in the troubled cell defined by

Q i = 1

|Ωe|∫Ωe

Q idx. (23)

For systems, we perform the limiting strategy in the local characteristic space. In x-direction, in the troubled cell we find the matrix R and its inverse R−1 which are used to diagonalize the Jacobian matrix evaluated at the mean conservative variables. To transform all the polynomials to the characteristic fields by multiplying R−1 to these polynomials:

Θ0 = R−1 Q 0, Θ i = R−1 Q i, (24)

and the limited approximate solution (20) can be rewritten as

Θminj ≤ Θ0, j = (R−1 Q 0

)j + α j

(R−1δ Q 0

)j = Θ0, j + α jδΘ0, j ≤ Θmax

j . (25)

For the j-th variable, the weighted minimum and maximum values are defined by

Θminj = ω0 min

Ωe(Θ0, j) +

∑ωi min

Ωe(Θi, j), Θmax

j = ω0 maxΩe

(Θ0, j) +∑

ωi maxΩe

(Θi, j), (26)

180 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

where the normalized nonlinear weights are

ω0 = ω0

ω0 +∑k ωk, ωi = ωi

ω0 +∑k ωk, (27)

ω0 = γ0

(ε + �0)2, ωi = γi

(ε + �i)2, ε = 10−6, (28)

γ0 = 0.99, γi = 0.01|ni0.x|∑k |nk0,x| , �0 =

∫Ωe

|δΘ0, j|2dx, �i =∫Ωe

|δΘi, j|2dx, (29)

where ni0.x is the x-component of the normalized cell barycenter vector:

�ni0 = xc,0 − xc,i

|xc,0 − xc,i| , (30)

in which xc,0 is the cell barycenter of the troubled cell and xc,i is the barycenter of the i-th adjacent cell.Then the parameter α j can be obtained as

α j = min(αmin

j ,αmaxj

), (31)

αminj =

⎧⎨⎩Φ(

Θminj −Θ0, j

minΩe (Θ0, j)−Θ0, j), minΩe (Θ0, j) − Θ0, j = 0,

1, else,

(32)

αmaxj =

⎧⎨⎩Φ(

Θmaxj −Θ0, j

maxΩe (Θ0, j)−Θ0, j), maxΩe (Θ0, j) − Θ0, j = 0,

1, else.

(33)

Barth and Jespersen [17] used a non-differentiable function Φ(y) = min(1, y) for the calculation of the parameter α j . However, the non-differentiability causes degradation in convergence performance [20]. In the current approach, a smooth function is used as

Φ(y) = erf(1.2y8 + √

π y/2), (34)

where erf (y) is the error function and it is a good approximation of the function min(1, y).Finally, the results are transformed back to the original space by multiplying R from left side as

Qx0 = RΘ0. (35)

In y and z directions, we can also obtain the limited approximations Qy0 and Q

z0. The final approximation in the

troubled cell is defined as the arithmetic mean of these approximations in different directions. Besides, we should maintain the density and pressure positive by using the method proposed by Zhang et al. [21].

In the original DG methods, different schemes are used for the inviscid flux and viscous flux calculation. Due to the relationship between the distribution function f and the fluxes, we can directly calculate these fluxes in Eqs. (13) and (15)as follows once the time-dependent distribution function f is determined,

∫Ωe

( �t∫0

�F dt

)• ∇υ j(x)dx =

∫Ωe

( �t∫0

∫�uψ f (x, t, �u, ξ)dΞdt

)• ∇υ j(x)dx, j = 1, · · · , l − 1, (36)

∫∂Ωe

( �t∫0

�F dt

)• �nυ j(x)dS =

∫∂Ωe

( �t∫0

∫�uψ f (x, t, �u, ξ)dΞdt

)• �nυ j(x)dS, j = 0,1, · · · , l − 1. (37)

In the following, we are going to present the distribution function evaluation inside the cell and at the cell interface.

2.2. Gas distribution function inside a cell

For the DG method, the macroscopic flow variables inside a cell are updated automatically with smooth assumption. Here the smooth distribution of the flow variables in a cell is represented by a 3rd order Taylor series expansion at the barycenter of the cell (subscript or superscript “c” indicates the barycenter of the cell):

Q h(x, t) = Q c(t) + ∂ Q c(t)

∂xυ ′′

1 + ∂ Q c(t)

∂ yυ ′′

2 + ∂ Q c(t)

∂zυ ′′

3 + ∂2 Q c(t)

∂x∂ yυ ′′

4 + ∂2 Q c(t)

∂x∂zυ ′′

5

+ ∂2 Q c(t)

∂ y∂zυ ′′

6 + ∂2 Q c(t)

∂x2υ ′′

7 + ∂2 Q c(t)

∂ y2υ ′′

8 + ∂2 Q c(t)

∂z2υ ′′

9 , (38)

with υ ′′ = x − xc, υ ′′ = y − yc, υ ′′ = z − zc, υ ′′ = υ ′′υ ′′, υ ′′ = υ ′′υ ′′, υ ′′ = υ ′′υ ′′, υ ′′ = 1 υ ′′υ ′′, υ ′′ = 1 υ ′′υ ′′, υ ′′ = 1 υ ′′υ ′′ .

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 181

According to Eq. (3), we can obtain the corresponding gas distribution function,

f (x, t, u, v, w, ξ) = f c(t) + ∂ f c(t)

∂xυ ′′

1 + ∂ f c(t)

∂ yυ ′′

2 + ∂ f c(t)

∂zυ ′′

3 + ∂2 f c(t)

∂x∂ yυ ′′

4 + ∂2 f c(t)

∂x∂zυ ′′

5

+ ∂2 f c(t)

∂ y∂zυ ′′

6 + ∂2 f c(t)

∂x2υ ′′

7 + ∂2 f c(t)

∂ y2υ ′′

8 + ∂2 f c(t)

∂z2υ ′′

9 . (39)

Based on the Chapman–Enskog expansion (6), a 3rd order NS gas distribution function f around t = 0 at any point(x, y, z) inside the cell has the form

f (x, t, u, v, w, ξ) = f I + f V , (40)

f I = gc0(dc00 + dc0

1 υ ′′1 + dc0

2 υ ′′2 + dc0

3 υ ′′3 + bc0

xyυ′′4 + bc0

xzυ′′5 + bc0

yzυ′′6 + bc0

xxυ′′7 + bc0

yyυ′′8 + bc0

zzυ′′9

), (41)

f V = −τ gc0(Dc00 + Dc0

1 υ ′′1 + Dc0

2 υ ′′2 + Dc0

3 υ ′′3

), (42)

with the notations:

gc0 = g(xc, yc, zc,0, u, v, w, ξ),

Bc00 = uac0

x + vac0y + wac0

z , Bc01 = Bc0

0 + Ac0t , Bc0

2 = ubc0xt + vbc0

yt + wbc0zt , Bc0

3 = Bc02 + bc0

tt ,

Bc04 = ubc0

xx + vbc0xy + wbc0

xz , Bc05 = ubc0

xy + vbc0yy + wbc0

yz, Bc06 = ubc0

xz + vbc0yz + wbc0

zz ,

Dc00 = Bc0

1 + Bc03 t, Dc0

1 = Bc04 + bc0

xt , Dc02 = Bc0

5 + bc0yt, Dc0

3 = Bc06 + bc0

zt ,

dc00 = 1 + Ac0

t t + 12 bc0

tt t2, dc01 = ac0

x + bc0xt t, dc0

2 = ac0y + bc0

ytt, dc03 = ac0

z + bc0zt t,

bc0xt = ac0

x Ac0t + Ac0

xt , bc0yt = ac0

y Ac0t + Ac0

yt, bc0zt = ac0

z Ac0t + Ac0

zt , bc0tt = Ac0

t Ac0t + Ac0

tt ,

bc0xy = ac0

x ac0y + ac0

xy, bc0xz = ac0

x ac0z + ac0

xz , bc0yz = ac0

y ac0z + ac0

yz,

bc0xx = ac0

x ac0x + ac0

xx, bc0yy = ac0

y ac0y + ac0

yy, bc0zz = ac0

z ac0z + ac0

zz ,

ac0x = 1

gc0∂ gc0

∂x , ac0y = 1

gc0∂ gc0

∂ y , ac0z = 1

gc0∂ gc0

∂z , Ac0t = 1

gc0∂ gc0

∂t ,

ac0xx = ∂ac0

x∂x , ac0

yy = ∂ac0y

∂ y , ac0zz = ∂ac0

z∂z , Ac0

tt = ∂ Ac0t

∂t ,

ac0xy = ∂ac0

x∂ y = ∂ac0

y∂x , ac0

xz = ∂ac0x

∂z = ∂ac0z

∂x , ac0yz = ∂ac0

y∂z = ∂ac0

z∂ y ,

Ac0xt = ∂ac0

x∂t = ∂ Ac0

t∂x , Ac0

yt = ∂ac0y

∂t = ∂ Ac0t

∂ y , Ac0zt = ∂ac0

z∂t = ∂ Ac0

t∂z .

The physical collision time τ is related to the dynamical viscosity coefficient μc0 and the pressure pc0 at the cell barycenter:

τ = μc0

pc0. (43)

In the above formulations, the equilibrium state gc0 is determined from the conservative flow variables at the cell barycenter and other expansion coefficient φ can be always written as follows [13]:

φ = φ1 + φ2u + φ3 v + φ4 w + 1

2φ5(u2 + v2 + w2 + ξ2)= (φ1, φ2, φ3, φ4, φ5) • ψ . (44)

Introducing the following notation

〈· · ·〉 =∫

gc0(· · ·)ψdΞ, (45)

then all coefficients can be obtained by the following equations which are directly derived from the conservation constraints equation (5),

⟨ac0

x

⟩= ∂ Q c(0)

∂x→ (

ac0x,1,ac0

x,2,ac0x,3,ac0

x,4,ac0x,5

),

⟨ac0

y

⟩= ∂ Q c(0)

∂ y→ (

ac0y,1,ac0

y,2,ac0y,3,ac0

y,4,ac0y,5

),

⟨ac0

z

⟩= ∂ Q c(0)

∂z→ (

ac0z,1,ac0

z,2,ac0z,3,ac0

z,4,ac0z,5

),

⟨uac0

x + vac0y + wac0

z + Ac0t

⟩= 0 → (Ac0

t,1, Ac0t,2, Ac0

t,3, Ac0t,4, Ac0

t,5

),

⟨ac0

xx + ac0x ac0

x

⟩= ∂2 Q c(0)

2→ (

ac0xx,1,ac0

xx,2,ac0xx,3,ac0

xx,4,ac0xx,5

),

∂x

182 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

⟨ac0

yy + ac0y ac0

y

⟩= ∂2 Q c(0)

∂ y2→ (

ac0yy,1,ac0

yy,2,ac0yy,3,ac0

yy,4,ac0yy,5

),

⟨ac0

zz + ac0z ac0

z

⟩= ∂2 Q c(0)

∂z2→ (

ac0zz,1,ac0

zz,2,ac0zz,3,ac0

zz,4,ac0zz,5

),

⟨ac0

xy + ac0x ac0

y

⟩= ∂2 Q c(0)

∂x∂ y→ (

ac0xy,1,ac0

xy,2,ac0xy,3,ac0

xy,4,ac0xy,5

),

⟨ac0

xz + ac0x ac0

z

⟩= ∂2 Q c(0)

∂x∂z→ (

ac0xz,1,ac0

xz,2,ac0xz,3,ac0

xz,4,ac0xz,5

),

⟨ac0

yz + ac0y ac0

z

⟩= ∂2 Q c(0)

∂ y∂z→ (

ac0yz,1,ac0

yz,2,ac0yz,3,ac0

yz,4,ac0yz,5

),⟨

u(ac0

xx + ac0x ac0

x

)+ v(ac0

xy + ac0x ac0

y

)+ w(ac0

xz + ac0x ac0

z

)+ Ac0xt + ac0

x Ac0t

⟩= 0 → (Ac0

xt,1, Ac0xt,2, Ac0

xt,3, Ac0xt,4, Ac0

xt,5

),⟨

u(ac0

xy + ac0x ac0

y

)+ v(ac0

yy + ac0y ac0

y

)+ w(ac0

yz + ac0y ac0

z

)+ Ac0yt + ac0

y Ac0t

⟩= 0 → (Ac0

yt,1, Ac0yt,2, Ac0

yt,3, Ac0yt,4, Ac0

yt,5

),⟨

u(ac0

xz + ac0x ac0

z

)+ v(ac0

yz + ac0y ac0

z

)+ w(ac0

zz + ac0z ac0

z

)+ Ac0zt + ac0

z Ac0t

⟩= 0 → (Ac0

zt,1, Ac0zt,2, Ac0

zt,3, Ac0zt,4, Ac0

zt,5

),⟨

u(

Ac0xt + ac0

x Ac0t

)+ v(

Ac0yt + ac0

y Ac0t

)+ w(

Ac0zt + ac0

z Ac0t

)+ Ac0tt + Ac0

t Ac0t

⟩= 0 → (Ac0

tt,1, Ac0tt,2, Ac0

tt,3, Ac0tt,4, Ac0

tt,5

).

Therefore, the integration equation (36) inside each cell can be calculated explicitly as:

∫Ωe

( �t∫0

�F dt

)• ∇v j(x)dx = �F 0 •

∫Ωe

∇υ j(x)dx +9∑

i=1

�F i •∫Ωe

υ ′′i ∇υ j(x)dx, (46)

in which⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�F 0 = C1〈�u〉 + C2〈�u Ac0t 〉 − C3〈�uBc0

0 〉 − C4〈�uBc02 〉 + C5〈�ubc0

tt 〉,�F 1 = C1〈�uac0

x 〉 + C2〈�ubc0xt 〉 − C3〈�uBc0

4 〉,�F 2 = C1〈�uac0

y 〉 + C2〈�ubc0yt〉 − C3〈�uBc0

5 〉,�F 3 = C1〈�uac0

z 〉 + C2〈�ubc0zt 〉 − C3〈�uBc0

6 〉,�F 4 = C1〈�ubc0

xy〉, �F 5 = C1〈�ubc0xz〉, �F 6 = C1〈�ubc0

yz〉,�F 7 = C1〈�ubc0

xx〉, �F 8 = C1〈�ubc0yy〉, �F 9 = C1〈�ubc0

zz〉,C1 = �t, C2 = �t(�t

2 − τ ), C3 = �tτ , C4 = τ�t2

2 , C5 = �t3−3τ�t2

6 .

(47)

2.3. Gas distribution function at a cell interface

In the general case, a discontinuous initial condition will appear at the cell interface, which corresponds to the non-equilibrium state at both sides of the cell interface. Starting from the initial discontinuity, a smooth continuous equilibrium state will be formed from the initial non-equilibrium state in the gas evolution process through particle collisions. The equilibrium state across a cell interface will be constructed through the local Linear Least Square method. The real gas distribution function at a cell interface depends closely on the intensity of particle collisions, which is basically a hybrid function of the equilibrium and non-equilibrium states.

In the general discontinuous case, the gas discontinuous distribution function f can be evaluated based on the integral solution of the gas kinetic BGK model equation (1) [13,14]:

f (x, t, u, v, w, ξ) = 1

τ

t∫0

g(x′, y′, z′, t′, u, v, w, ξ

)e(t′−t)/τ dt′ + e−t/τ f0(x − ut, y − vt, z − wt,0, u, v, w, ξ) (48)

where (x′, y′, z′) = (x − u(t − t′), y − v(t − t′), z − w(t − t′)) is the particle trajectory, t is the time, (x, y, z) is the point at the cell interface, and f0 is the initial gas distribution function at the beginning of each time step (t = 0). In order to figure out the gas distribution function at the cell interface, two unknowns, g and f0 in the above equation have to be specified. Due to a discontinuity at the cell interface, see Fig. 1, the gas distribution functions will be different on different sides of a cell interface. Therefore, the initial gas distribution function f0 can be constructed as

f0(x,0, u, v, w, ξ) ={

f L0 (x,0, u, v, w, ξ), un > 0,

f R0 (x,0, u, v, w, ξ), un < 0,

(49)

where un = �u • �n is the velocity in the normal direction of the interface.

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 183

Fig. 1. A cell interface sketch.

Since the expansion point is at interface barycenter (x f c, y f c, z f c), a 3rd-order Taylor expansion of the initial gas distri-bution function f0 becomes

f0(x,0, u, v, w, ξ) = f f c0 + ∂ f f c0

∂xυ ′′′

1 + ∂ f f c0

∂ yυ ′′′

2 + ∂ f f c0

∂zυ ′′′

3 + ∂2 f f c0

∂x∂ yυ ′′′

4 + ∂2 f f c0

∂x∂zυ ′′′

5

+ ∂2 f f c0

∂ y∂zυ ′′′

6 + ∂2 f f c0

∂x2υ ′′′

7 + ∂2 f f c0

∂ y2υ ′′′

8 + ∂2 f f c0

∂z2υ ′′′

9 , (50)

with υ ′′′1 = x − x f c, υ ′′′

2 = y − y f c, υ ′′′3 = z − z f c, υ ′′′

4 = υ ′′′1 υ ′′′

2 , υ ′′′5 = υ ′′′

1 υ ′′′3 , υ ′′′

6 = υ ′′′2 υ ′′′

3 , υ ′′′7 = 1

2 υ ′′′ 21 , υ ′′′

8 = 12 υ ′′′ 2

2 ,

υ ′′′9 = 1

2 υ ′′′ 23 .

Substituting Eq. (6) into the above equation we have

f0(x, y, z,0, u, v, w, ξ) = g f c0

⎡⎢⎢⎣

1 + a f c0x υ ′′′

1 + a f c0y υ ′′′

2 + a f c0z υ ′′′

3 + b f c0xy υ ′′′

4

+ b f c0xz υ ′′′

5 + b f c0yz υ ′′′

6 + b f c0xx υ ′′′

7 + b f c0yy υ ′′′

8 + b f c0zz υ ′′′

9

− τ (B f c01 + D f c0

1 υ ′′′1 + D f c0

2 υ ′′′2 + D f c0

3 υ ′′′3 )

⎤⎥⎥⎦ . (51)

Since there are two different reconstructed distribution functions at both sides of a cell interface, the initial gas distribu-tion function e−t/τ f0 in Eq. (48) by using Eq. (51) becomes,

e−t/τ f0(x − ut, y − vt, z − wt,0, u, v, w, ξ) = e−t/τn f L0 H[un] + e−t/τn f R

0

(1 − H[un]

), (52)

in which

f ∗0 = g∗, f c0

⎡⎢⎢⎢⎢⎢⎣

1 − t B∗ f c00 − τ B∗ f c0

1 + τ t B∗ f c02 + (τ t + 1

2 t2)(uB∗ f c04 + v B∗ f c0

5 + w B∗ f c06 )

+ (a∗ f c0x − t B∗ f c0

4 − τ D∗ f c01 )υ ′′′

1 + (a∗ f c0y − t B∗ f c0

5 − τ D∗ f c02 )υ ′′′

2

+ (a∗ f c0z − t B∗ f c0

6 − τ D∗ f c03 )υ ′′′

3

+ b∗ f c0xy υ ′′′

4 + b∗ f c0xz υ ′′′

5 + b∗ f c0yz υ ′′′

6 + b∗ f c0xx υ ′′′

7 + b∗ f c0yy υ ′′′

8 + b∗ f c0zz υ ′′′

9

⎤⎥⎥⎥⎥⎥⎦ , (53)

where H[x] is the Heaviside function defined as

H[x] ={

0, x < 0

1, x > 0,(54)

and “∗” can take “R” or “L” for the right and left sides states.With the initial distribution of macroscopic flow variables at the left and right sides of a cell interface, which are given

by the DG method initially, we can uniquely determine the Maxwellian distributions g R, f c0 and gL, f c0 first, and obtain other expansion coefficients by using the same method presented in Section 2.2.

For the modeling of the local equilibrium state g in Eq. (48) across a cell interface, the Taylor expansion can be used for its construction

184 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Fig. 2. Gauss Quadrature Points distribution.

g(x, y, z, t, u, v, w, ξ)

= g f c0[d f c00 + d f c0

1 υ ′′′1 + d f c0

2 υ ′′′2 + d f c0

3 υ ′′′3 + b f c0

xy υ ′′′4 + b f c0

xz υ ′′′5 + b f c0

yz υ ′′′6 + b f c0

xx υ ′′′7 + b f c0

yy υ ′′′8 + b f c0

zz υ ′′′9

].

(55)

Substituting Eq. (55) into Eq. (48), we can get the local equilibrium state integration

1

τ

t∫0

g(x′, y′, z′, t′, u, v, w, ξ

)e(t′−t)/τ dt′

= g f c0

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

C1 + C3 B f c00 − C4(uB f c0

4 + v B f c05 + w B f c0

6 ) + C5 A f c0t − C6 B f c0

2 + C7b f c0tt

+ (C1a f c0x + C3 B f c0

4 + C5b f c0xt )υ ′′′

1 + (C1a f c0y + C3 B f c0

5 + C5b f c0yt )υ ′′′

2

+ (C1a f c0z + C3 B f c0

6 + C5b f c0zt )υ ′′′

3

+ C1b f c0xy υ ′′′

4 + C1b f c0xz υ ′′′

5 + C1b f c0yz υ ′′′

6 + C1b f c0xx υ ′′′

7 + C1b f c0yy υ ′′′

8 + C1b f c0zz υ ′′′

9

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

, (56)

with C1 = 1 − e−t/τn , C2 = τC1, C3 = te−t/τn − C2, C4 = ( 12 t + τ )te−t/τn , C5 = t − C2, C6 = tC2, C7 = 1

2 t2 − τ t .Here two collision times [14] τ and τn are used. The numerical collision time τn controls the contributions from f0 and

g in the final integral solution in the discontinuous case. Therefore, the numerical collision time τn is modeled with the consideration of the cell size and artificial discontinuous jump

τn = μ f c0

p f c0+ β

∣∣(xRc − xL

c

) • �n∣∣√λ f c0 |pR, f c0 − pL, f c0||pR, f c0 + pL, f c0| , (57)

in which xRc and xL

c are the coordinates vectors of the right and left sides cell barycenters.The physical collision time τ represents the physical viscous and heat conducting effect and has the same definition as

Eq. (43) by using the variables μ f c0 and p f c0.In order to fully determine the above equilibrium state integration, a smooth distribution of flow variables around a cell

interface is needed for the equilibrium state construction. Due to the solution discontinuity at the cell interface, an LLSmethod will be used to reconstruct the smooth distribution of flow variables across the cell interface for the evaluation of the Maxwellian distribution g f c0 and other expansion coefficients in Eq. (56). Finally combining Eq. (52) and (56) together, we get the time dependent distribution function along the cell interface. In Section 2.4, we will present the LLS method for the cross interface flow variable construction.

2.4. The Linear Least Squares method

For any cell interface, there are two neighboring cells, as shown in Fig. 1. Due to the possible discontinuity at the cell interface, the conservative variables may take jumps at the cell interface. Firstly, we introduce the Gauss Quadrature Points used here. For the 3rd order cases, the Gauss Quadrature Points (Red Points) distribution on the interface is shown in Fig. 2.

At the ith Gaussian Quadrature Point, with the right side and left side conservative variable vectors Q Rg,i and Q Lg,i , we can define the corresponding Maxwellian distributions g Rg,i and gLg,i . Then, the conservative variable vector Q g,i can be constructed from [13]

Q g,i =∫

g g,iψdΞ =∫

u <0

g Rg,iψdΞ +∫

u >0

gLg,iψdΞ. (58)

n,i n,i

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 185

Similarly, the conservative variable vector Q f c on the interface barycenter becomes

Q f c =∫

g f cdΞ =∫

un, f c<0

g R f cψdΞ +∫

un, f c>0

gLf cψdΞ. (59)

At the same time, the conservative variable vectors Q Rc and Q Lc on the barycenter of the two neighboring cells are known. Based on these available values, a smooth distribution of the conservative variables around the cell interface can be gotten using a Linear Least Square method.

The smooth distribution of the flow variables across the interface can be written as

Q h − Q f c =9∑

i=1

Q iυ f ,i, (60)

where the basis functions are defined as

υ f ,1 = υ ′′′1

�x f c, υ f ,2 = υ ′′′

2

�y f c, υ f ,3 = υ ′′′

3

�z f c, υ f ,4 = υ f ,1υ f ,2, υ f ,5 = υ f ,1υ f ,3,

υ f ,6 = υ f ,2υ f ,3, υ f ,7 = (υ f ,1)2

2, υ f ,8 = (υ f ,2)

2

2, υ f ,9 = (υ f ,3)

2

2,

�x f c = max(�xL,�xR), �y f c = max(�yL,�yR), �z f c = max(�zL,�zR).

To determine these unknowns Q = (Q 1, Q 2, · · · , Q 9) in Eq. (60), we solve the following Linear Least Squares problem:

minimizeQ

‖b − A Q ‖2 and minimizeQ

| Q |2, (61)

where

b = [Q Rc − Q f c, Q Lc − Q f c, Q g,1 − Q f c, · · · , Q g,q − Q f c]T,

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

υRcf ,1 υRc

f ,2 · · · υRcf ,9

υ Lcf ,1 υ Lc

f ,2 · · · υ Lcf ,9

υg,1f ,1 υ

g,1f ,1 · · · υ

g,1f ,1

υg,2f ,1 υ

g,2f ,2 · · · υ

g,2f ,9

......

. . ....

υg,qf ,1 υ

g,qf ,2 · · · υ

g,qf ,9

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

in which q is the number of the Gaussian Quadrature Points, the superscript “Rc” means the right hand side cell barycenter, “Lc” the left hand side cell barycenter, and “g , i” is the ith Gaussian quadrature point along the interface.

Finally, the Maxwellian distribution g f c0 and all other coefficients in Eq. (56) can be fully determined.

Remarks.

(1) In the gas-kinetic BGK scheme, some moments of a Maxwellian distribution function with bounded and unbounded integration limits are needed, and their general formulas can be found in [13].

(2) The current method integrates the governing equations in time explicitly, same as that in [12,14].(3) The method is a multi-dimensional BGK scheme, which not only couples the convective and dissipative terms, but also

includes both discontinuous and continuous flow evolution in the flux evaluation at a cell interface through a relaxation process of the gas distribution function. An analytic time and space dependent gas distribution function is used to calculate the flux both inside volume and at the cell interface. The flux can be directly integrated in space and time without applying the integration method with the Gaussian Quadrature Points. This is useful for saving CPU cost.

3. Numerical examples

The multi-dimensional high-order gas-kinetic DG method will be tested in a few examples in this section.

3.1. Accuracy test – advection of density perturbation

A quasi-two-dimensional Euler problem with exact solution is used for the accuracy test. The physical collision time τ is set to be zero to give the Euler solution. The computational domain Ω is [0, 1] × [0, 1] × [0, 0.1] ⊂ R

3 and the initial flow condition is

186 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Fig. 3. The sample grid with 10 cells in x and y direction, and 1 cell in z direction.

Fig. 4. L1 errors and orders of convergence (left) and L∞ errors and the orders of convergence (right).

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ρ(x, y, z,0) = 1 + 0.2 sin(2π(x + y)

),

U (x, y, z,0) = 0.7,

V (x, y, z,0) = 0.3,

W (x, y, z,0) = 0,

p(x, y, z,0) = 1.

(62)

Periodic boundary conditions are applied in both x and y directions, and the symmetry boundary condition is applied for z direction. The computation is up to t = 1 and the exact density is ρ(x, y, z, t) = 1 + 0.2 sin(2π(x + y − t)). A sample uniform grid is used here for the calculation, see Fig. 3. The grid in the x–y plane is refined by quartering all cells in the mesh refinement study, where only one layer of grid in the z direction is kept. The spatial errors and the orders of convergence are shown in Fig. 4. The results show that the method is indeed a 3rd order accuracy for both L1 and L∞norms.

3.2. Accuracy test – Taylor vortex

This is one of the test cases with the exact solution as well. It has been used for the accuracy test in [22]. The computa-tional domain Ω is [0, 1] × [0, 1] × [0, 0.1] ⊂ R

3 and the exact solution is

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 187

Fig. 5. L1 errors and orders of convergence (left) and L∞ errors and the orders of convergence (right).⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(x, y, z, t) = 1.0;U (x, y, z, t) = − 1

2πcos(2πx) sin(2π y)e−8π2νt,

V (x, y, z, t) = 1

2πsin(2πx) cos(2π y)e−8π2νt,

W (x, y, z, t) = 0,

p(x, y, z, t) = 1 − 1

4

[cos(4πx)

4π2+ cos(4π y)

4π2

]e−16π2νt,

(63)

where ν = 0.001 is the shear viscosity. The initial condition can be gotten by setting t in Eq. (63) to be zero. Periodic boundary conditions are applied in both x and y directions, and the symmetry boundary condition is applied in the zdirection. Actually, a source term should be added to the equation of energy to maintain the energy conservation. The grid is similar with the one in Fig. 3. It is refined in the x–y plane by quartering all cells in the mesh refinement study, where only one layer of grid in the z direction is kept. The error of the U velocity is shown in Fig. 5. The results show that the method is indeed a 3rd order accuracy for both L1 and L∞ norms.

3.3. Double Mach reflection problem

The problem was proposed in [23]. A Mach 10 shock in a perfect gas with the specific heat ratio γ = 1.4 is initially set up with an inclined angle 60◦ hitting on the wall, which is defined as the x-axis. The computational domain Ω is [0, 3.5] × [0, 1] × [0, 0.1] ⊂ R

3. The cell size using here is h = 1/100 in x and y direction and 0.1 in z direction, leading to 350 × 100 × 1 computational cells. The left x boundary is an inflow condition and the right x boundary is an outflow condition. The reflecting wall boundary condition is adopted on the lower y boundary for x ≥ 0.2 and the rest boundary condition at bottom is set by the initial post-shock condition. The exact solution of an isolated moving oblique shock wave with shock Mach number Ms = 10 is imposed on the upper y boundary. By using the Rankine–Hugoniot conditions, we can get the initial condition in front of and behind the shock wave

(ρ, u, v, w, p) ={

(8.0,8.25 cos(30◦),−8.25 sin(30◦),0.0,116.5), x < 0.2 + y tan(30◦),(1.4,0.0,0.0,0.0,1.0), x ≥ 0.2 + y tan(30◦).

(64)

Fig. 6 and Fig. 7 show the density contours at t = 0.2 with 30 isolines range from 1.3965 to 22.682. They are comparable with the results in [1].

3.4. The forward facing step problem

The problem was also proposed by Woodward and Colella in [23] and has become a well-known benchmark problem for a compressible gas. The wind tunnel has 1 unit high and 3 unit long. The step height is 0.2 units and located 0.6 units from the left-hand end of the tunnel. A uniform right-going flow with Ma = 3 is initialized for the problem and the air condition is

(ρ, u, v, w, p) = (1.4,3.0,0.0,0.0,1.0). (65)

The reflecting wall boundary condition is used for the upper and lower y boundaries. Constant values are imposed on the left x boundary and the extrapolation is applied on the right x boundary, respectively. The symmetry boundary condition

188 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Fig. 6. Density contours obtained at t = 0.2. The 3rd traditional DG method by Cockburn and Shu [1] with the mesh size h = 1/120 (top). The 3rd gas-kinetic DG method with the mesh size h = 1/100 (bottom). Here 30 isolines ranging from 1.3965 to 22.682 are plotted.

Fig. 7. Density contours obtained at t = 0.2 using the 3rd gas-kinetic DG method—Zoom in the interaction zone with 30 isolines ranging from 1.3965 to 22.682.

Fig. 8. Meshes used for Forward Facing Step problem. The left one is a coarse mesh with the minimum mesh size h = 0.005 near the step corner and the maximum size h = 0.025 on other boundaries. The right is a fine mesh with the minimum mesh size h = 0.002 near the step corner and the maximum size h = 0.01 on other boundaries.

is applied for z direction. Two non-uniform meshes are used and shown in Fig. 8. The maximum mesh size is h = 0.025 and h = 0.01, respectively, and the mesh is refined near the step corner to reduce the artifacts caused by the singular point (the step corner) [1]. One layer of grid is used in the z direction. Fig. 9 shows the density contours at t = 4.0 with 30 isolines ranging from 0.09 to 6.24, and they are compared with the results from Ref. [1]. It shows good performance of the current scheme.

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 189

Fig. 9. Density contours at t = 4 with 30 isolines ranging from 0.09 to 6.24. The 3rd traditional DG method by Cockburn and Shu [1] with the mesh size h = 1/40 and 1/80 (top). The 3rd gas kinetic DG method with the mesh size h = 1/40 and 1/100 (bottom).

3.5. Viscous test case

Shu et al. [6,24,25] have pointed out that a naive generalization of the DG method to a convection–diffusion problem containing higher spatial derivatives may present unphysical solution. The numerical results have shown that the appli-cation of the traditional DG method directly to the heat equation containing a second derivative could yield a method which behaves nicely in the computation, but is weakly unstable and has O (1) errors to the exact solution. This is called the “non-conforming problem”, see Fig. 10 [6]. Generally, these equations with higher order derivatives in traditional DG methods have to be rewritten into a first order system, which is used to obtain a stable and accurate solution. Here, the gas-kinetic DG method solves the viscous and heat conducting flows directly using a single gas distribution function for all flux evaluations. First, we use a special test case to check if the numerical solution is converging to the exact solution. The NS equations with source terms are designed as

∂ Q (x, t)

∂t+ ∇ • �F = S, (66)

S = [0,0.4πρR(e− 8π2γμ

ρ Pr t) cos[2π(x + y − γ t)

],0.4πρR

(e− 8π2γμ

ρ Pr t) cos[2π(x + y − γ t)

],0,0

], (67)

where R denotes the specific gas constant, Pr is the Prandtl number. The source terms in the 2nd and 3rd equations are added to maintain the constant flow velocities.

The initial flow condition is

[ρ, U , V , W , T ] = [1,0.7,0.3,0,1 + 0.2 sin[2π(x + y)

]]. (68)

The exact solution of the temperature is T (x, y, z, t) = 1 + 0.2(e− 8π2γμρ Pr t

) sin[2π(x + y − γ t)].The grid is similar with the one in Fig. 3, with 10 or 40 cells in both x and y direction, and 1 layer of grid in the

z direction. The computation is up to t = 0.7 and the results, shown in Fig. 11, indicate that the numerical solution can converge to the exact solution with the mesh refinement, where the non-conforming problem is not present from the current scheme.

3.6. Hypersonic laminar flow past a circular cylinder

This test case is taken from the experiment done by Wieting [26]. The flow condition is given as M∞ = 8.03, T∞ =124.94 K for the far field, the wall temperature T W = 294.44 K, and the Reynolds number is Re = 1.835 × 105 with the cylinder radius and the far field flow parameters. A non-uniform grid of 80 ×160 ×1 mesh points is used with the near-wall cell width of 5E–5. Luo et al. [10] has used this case to demonstrate the robustness of the numerical method for accurate and reliable prediction of the heat flux in the hypersonic regime. Fig. 12 and Fig. 13 show the numerical results, which have good agreement with the experimental data.

190 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Fig. 10. Reproduced from [6]. The application of the 3rd traditional DG method directly to the heat equation ut − uxx = 0 containing a second derivative with an initial condition u(x, 0) = sin(x). The results are obtained at t = 0.7. A third order Runge–Kutta time stepping method is used for time accuracy.

Fig. 11. The temperature plot along the line y = x.

3.7. 3D Lid-driven cavity flow with different Reynolds numbers

The lid-driven cavity is a classic benchmark laminar incompressible flow. In the recent years, the lid-driven cavity prob-lem has been used as a standard Reynolds number dependent benchmark test for 3D computational fluid dynamics (CFD) methods [27,28]. The 3D cavity is a unit cube, with the top wall moving in the positive x-axis with unit velocity, U = 1. The non-slip boundary condition is applied for other walls. The initial condition is static everywhere. The top plane velocity gen-erates vorticity which propagates throughout the domain until the flow field reaches a steady state. In order to demonstrate the robustness of the current method for incompressible flow, steady state solutions for three different Reynolds numbers 100, 400 and 1000 are computed here. A non-uniform Cartesian grid (36 × 36 × 32) is selected, see Fig. 14. The results are compared with the data from [27,28], see Fig. 15. We can see that the results calculated by the gas-kinetic DG method with a coarser grid are in good agreement with the data from [27] and [28]. It indicates that the gas-kinetic DG method has the capability of obtaining an accurate solution for the problem with a low Mach number without using the preconditioning techniques. Besides, it also illustrates that a higher order method can use a much less mesh point for the same accurate solution. The most important purpose of this test is to show that the same gas kinetic DG scheme has not only the shock capturing property, but also be able to obtain the viscous solution accurately.

4. Conclusions

A high order gas-kinetic discontinuous Galerkin method for viscous flow computation is proposed. Different from the traditional DG methods for the Navier–Stokes equations, the current scheme adopts a multi-dimensional kinetic formulation to obtain both inviscid and viscous fluxes. A high-order space and time dependent flux function has been constructed, which can be directly integrated in space and time without using Gaussian Quadrature point integration method. In terms of a high-order DG scheme, the direct integration in space and time significantly reduces the computational cost. A Linear Least Square method is used to reconstruct a smooth distribution of the flow variables around the cell interface for the equilibrium state construction. Based on the same kinetic DG method, the numerical examples include compressible and

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 191

Fig. 12. Pressure and temperature contours obtained using the 3rd gas kinetic DG method.

Fig. 13. Comparison of the computed pressure and heat flux along the cylindrical surface with the experimental data.

Fig. 14. A no uniform grid for 3D lid-driven cavity problem.

192 X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193

Fig. 15. Comparison of the velocity u along the line (x = 0,0 ≤ y ≤ 1, z = 0).

incompressible, and viscous and inviscid tests. The numerical results demonstrate the robustness and accuracy of the current DG method.

In summary, the DG formulation seeks compact and high-order accuracy in the algorithm development. The additional degrees of freedom associated with the high-order formulation are evaluated through the weak form of the original govern-ing equations. These associated degrees of freedom are equivalent to the updating of cell averaged high-order derivatives of the flow variables. Theoretically, any update of derivative of flow variable is doubtful in the discontinuous case. This is one of the reasons why the detection of trouble cell and the use of limiter are so important in the DG method. The DG formulation has to be converged to the finite volume method with the update of cell averaged conservative flow variables only in the discontinuous case, such as the practice of many hybrid DG methods. The reason for the current gas-kinetic DG method being more robust than the conventional one based on the Riemann solution is due to the following reason. With the enhancement of the viscosity coefficient through the determination of the particle collision time, even with the detection of troubled cell and the use of limiter, the gas kinetic scheme still tries to resolve the discontinuous region with a numerically smooth shock structure, which is much compatible with the smoothness assumption underlying the DG weak formulation.

Acknowledgements

This work was supported by Hong Kong Research Grant Council (621011, 620813, 16211014) and National Natural Sci-ence Foundation of China (Grant Nos. 51136003, 51276093).

References

[1] B. Cockburn, C.-W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V, J. Comput. Phys. 141 (2) (1998) 199–224.[2] W.H. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos report LA-UR-73-479, 1973.[3] P. Lesaint, P.A. Raviart, On a Finite Element Method for Solving the Neutron Transport Equation, Academic Press, New York, 1974.[4] G. Chavent, G. Salzano, A finite-element method for the 1-D water flooding problem with gravity, J. Comput. Phys. 45 (3) (1982) 307–344.[5] B. Cockburn, C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework,

Math. Comput. 52 (186) (1989) 411–435.[6] Y. Xu, C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (1)

(2010) 1–46.[7] F. Bassi, S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations,

J. Comput. Phys. 131 (2) (1997) 267–279.[8] B. Cockburn, C.W. Shu, The local discontinuous Galerkin method for time-dependent convection–diffusion systems, SIAM J. Numer. Anal. 35 (6) (1998)

2440–2463.[9] H. Liu, K. Xu, A Runge–Kutta discontinuous Galerkin method for viscous flow equations, J. Comput. Phys. 224 (2) (2007) 1223–1242.

[10] H. Luo, L.Q. Luo, K. Xu, A discontinuous Galerkin method based on a BGK scheme for the Navier–Stokes equations on arbitrary grids, Adv. Appl. Math. Mech. 1 (3) (2009) 301–318.

[11] G. Ni, S. Jiang, K. Xu, A DGBGK scheme based on WENO limiters for viscous and inviscid flows, J. Comput. Phys. 227 (11) (2008) 5799–5815.[12] K. Xu, Discontinuous Galerkin BGK method for viscous flow equations: one-dimensional systems, SIAM J. Sci. Comput. 25 (6) (2004) 1941–1963.[13] K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys.

171 (1) (2001) 289–335.[14] J. Luo, K. Xu, A high-order multidimensional gas-kinetic scheme for hydrodynamic equations, Sci. China, Technol. Sci. 56 (10) (2013) 2370–2384.[15] H. Luo, J.D. Baum, R. Löhner, A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys.

227 (20) (2008) 8875–8893.[16] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J.E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic

conservation laws, Appl. Numer. Math. 48 (3) (2004) 323–338.[17] T. Barth, D. Jespersen, The design and application of upwind schemes on unstructured meshes, in: 27th Aerospace Sciences Meeting, American Institute

of Aeronautics and Astronautics, 1989.[18] D. Kuzmin, A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods, J. Comput. Appl. Math. 233 (12) (2010) 3077–3085.

X. Ren et al. / Journal of Computational Physics 292 (2015) 176–193 193

[19] D. Kuzmin, Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis, Int. J. Numer. Methods Fluids 71 (9) (2013) 1178–1190.

[20] M. Krzysztof, O.-G. Carl, Limiters for unstructured higher-order accurate solutions of the Euler equations, in: 46th AIAA Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics, 2008.

[21] X. Zhang, Y. Xia, C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput. 50 (1) (2012) 29–62.

[22] Z. Guo, K. Xu, R. Wang, Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case, Phys. Rev. E 88 (3) (2013) 033305.

[23] P. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54 (1) (1984) 115–173.[24] B. Cockburn, C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (3) (2001) 173–261.[25] C.-W. Shu, Different formulations of the discontinuous Galerkin method for the viscous terms, in: Z.-C. Shi, M. Mu, W.M. Xue, J. Zou (Eds.), Advances

in Scientific Computing, Science Press, 2001, pp. 144–155.[26] A.R. Wieting, Experimental study of shock wave interface heating on a cylindrical leading edge, No. NASA TM-100484 1987.[27] K.L. Wong, A.J. Baker, A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm, Int. J. Numer. Methods Fluids 38 (2)

(2002) 99–123.[28] S. Albensoeder, H.C. Kuhlmann, Accurate three-dimensional lid-driven cavity flow, J. Comput. Phys. 206 (2) (2005) 536–558.