-
HAL Id: hal-01960018https://hal.umontpellier.fr/hal-01960018
Submitted on 19 Dec 2018
HAL is a multi-disciplinary open accessarchive for the deposit
and dissemination of sci-entific research documents, whether they
are pub-lished or not. The documents may come fromteaching and
research institutions in France orabroad, or from public or private
research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt
et à la diffusion de documentsscientifiques de niveau recherche,
publiés ou non,émanant des établissements d’enseignement et
derecherche français ou étrangers, des laboratoirespublics ou
privés.
Distributed under a Creative Commons Attribution| 4.0
International License
Simulation of a time dependentadvection-reaction-diffusion
problem using operatorsplitting and discontinuous Galerkin methods
with
application to plant root growthEmilie Peynaud
To cite this version:Emilie Peynaud. Simulation of a time
dependent advection-reaction-diffusion problem using
operatorsplitting and discontinuous Galerkin methods with
application to plant root growth. BIOMATH,Biomath Forum, 2018, 7
(2), pp.1812037. �10.11145/j.biomath.2018.12.037�.
�hal-01960018�
https://hal.umontpellier.fr/hal-01960018http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://hal.archives-ouvertes.fr
-
www.biomathforum.org/biomath/index.php/biomath
ORIGINAL ARTICLE
Operator splitting and discontinuous Galerkinmethods for
advection-reaction-diffusion
problem. Application to plant root growthEmilie Peynaud
CIRAD, UMR AMAP, Yaoundé, CamerounAMAP, University of
Montpellier, CIRAD, CNRS, INRA, IRD, Montpellier, France
University of Yaoundé 1, National Advanced School of
Engineering, Yaoundé, [email protected]
Received: 19 September 2017, accepted: 3 December 2018,
published: 18 December 2018
Abstract—Motivated by the need of developingnumerical tools for
the simulation of plant rootgrowth, this article deals with the
numerical res-olution of the C-Root model. This model describesthe
dynamics of plant root apices in the soil andit consists in a time
dependent advection-reaction-diffusion equation whose unique
unknown is thedensity of apices. The work is focused on
theimplementation and validation of a suitable nu-merical method
for the resolution of the C-Rootmodel on unstructured meshes. The
model is solvedusing Discontinuous Galerkin (DG) finite
elementscombined with an operator splitting technique. Aftera brief
presentation of the numerical method, theimplementation of the
algorithm is validated in asimple test case, for which an analytic
expressionof the solution is known. Then, the issue of
thepositivity preservation is discussed. Finally, the DG-splitting
algorithm is applied to a more realistic rootsystem and the results
are discussed.
Keywords-Time dependent advection-reaction-diffusion; Operator
splitting; Discontinuous
Galerkin method; Plant root growth simulation;
I. INTRODUCTION
The article is devoted to the numerical mod-eling of plant root
growth. This work has beenoriginally motivated by the need of
developingnumerical tools for the simulation of plant
growthdynamics. Due to the difficulty of doing non-destructive
observations of the underground partof plants (that allow to do
long term studies of thedynamics of tree roots for example),
mathematicalmodels are achieving an essential role.
Severaltheoretical and numerical challenges arise in thefield of
the simulation of the dynamics of plantroots [48], [47], [38], [2],
[39]. The mathemat-ical description of plant root is not trivial,
dueto the presence of many interactions arising inthe rhizosphere
and also due to the diversity ofplant root types. Mathematical
models based onthe use of partial differential equations are
usefultools to simulate the evolution of root densities in
Copyright: c© 2018 Peynaud. This article is distributed under
the terms of the Creative Commons Attribution License (CCBY 4.0),
which permits unrestricted use, distribution, and reproduction in
any medium, provided the original author and sourceare
credited.Citation: Emilie Peynaud, Operator splitting and
discontinuous Galerkin methods foradvection-reaction-diffusion
problem. Application to plant root growth, Biomath 7 (2018),
1812037,http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 1 of
19
http://www.biomathforum.org/biomath/index.php/biomathhttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
space and time [43], [44], [44], [45], [46], [41],[40], [1].
This formalism facilitates the couplingwith physical models such as
water and nutrienttransports [42], [43], [44], [41], [49]. And the
com-putational time for the simulation of such modelsis not
dependent on the number of roots which isuseful for applications at
large scale. The C-Rootmodel [1] is a generic model of the dynamics
ofroot density growth. This model takes only oneunknown which is
related to root densities suchas the density of apices, root length
density orbiomass density. It has only three parameters. Themodel
is said to be generic in the sense that itcan apply to a wide
variety of root system types.The model consists in a single
time-dependentadvection-reaction-diffusion equation, and one ofthe
challenge is to numerically solve the equation.In [1] and [2] the
authors solved the problemwith the finite difference method on
Cartesianmesh grids combined with an operator splittingtechnique.
Unfortunately, Cartesian mesh grids donot allow easily to mesh
complex soil geometries.From the theoretical and computational
point ofview, Cartesian grids also lead to difficulties for
arigorous study and validation of the model. Thatis why this
article focus on the development andimplementation of a suitable
numerical methodfor the resolution of the C-Root model on
tri-angular mesh grids, that allow to mesh complexgeometries.
However, one of the main difficultiesin the C-root model is that
the advection anddiffusion terms are not always of the same orderof
magnitude. It depends on the phase of the rootsystem development
[2]. As a result, the propertiesof the equation may vary along the
simulation:it can be either close to a hyperbolic problem orclose
to a parabolic problem.
In a previous work [3], the use of the Discon-tinuous Galerkin
method has been implementedand validated. Indeed, the usual choice
of theclassical Lagrange finite element method suffersfrom a lack
of stability when the advection termis dominant [4]. For this
reason, we implementeda discontinuous Galerkin (DG) method for
boththe advection and diffusion terms. All the three
operators where solved simultaneously using thesame time
approximation scheme (θ-scheme).
However, as explained in [6], for multi-biophysic problems it is
not efficient to use thesame numerical scheme for the different
operatorsof the system. For example, we may want to usethe Euler
explicit scheme for the advection termand an Euler implicit scheme
for the diffusion.The operator splitting technique [7], [8] is a
wellknown alternative for the resolution of equationshaving a
multi-biophysic behaviour that allows theuse of different time
schemes for each operator ofthe equation. The idea of the splitting
technique isto split the problem into smaller and simpler partsof
the problem so that each part can be solvedby an efficient and
suitable time scheme. Thismethods has been used for a wide range of
applica-tions dealing with the advection-reaction-diffusionequation
[9]. Operator splitting techniques havebeen extensively used in
combination with finitedifference methods [10], [2], finite volume
meth-ods [11], [12] but also with Continuous Galerkinmethods [13],
[14], [15], [16], [17]. To the bestof my knowledge, only very few
articles dealwith the use of the operator splitting techniquein
combination with the discontinuous Galerkinapproximation [18],
[19], [20], [21]. In this paper,we present a new application of the
operatorsplitting technique combined with discontinuousfinite
elements.
The paper is structured as follows. In sectionII, the C-root
growth model [1], [2], [3] is brieflydescribed. An analysis is also
provided, where Ishowed the existence and uniqueness of a
positivereal solution. In section III, the splitting
operatortechnique is introduced and applied to the C-Rootmodel,
combined with the use of discontinuousGalerkin approximations. In
section IV, the algo-ritm is implemented and validated using a
simpletest case for which an analytic expression of thesolution is
known. As an application, I providesimulations of the development
of eucalyptus rootsin section V. Finally, the paper ends with a
con-clusion and further improvements.
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 2 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
II. THE MODEL
A. Modelling root growth with PDE: the C-Rootmodel
The C-Root model [1] was developed to sim-ulate the growth of
dense root networks, usuallycomposed of fine roots, with negligible
secondarythickening. As presented in [1], the unknown vari-able u
is the number of apices per unit volume,but it can also stand for
the density of fine rootbiomass. The soil is considered as a
subdomainof Rd (with d = 1, 2 or 3). It is assumed that Ωhas smooth
boundaries (Lipschitz boundaries) de-noted ∂Ω. The C-Root model
combines advection,diffusion and reaction, which aggregate the
mainbiological processes involved in root growth, suchas primary
growth, ramification and root death.The reaction operator gives the
quantity of apices(or root biomass) produced in time, whereas
ad-vection and diffusion operators spatially distributethe whole
apices (or biomass) in the domain.
The reaction operator describes the evolution intime of the root
biomass in a given domain. Inthe C-Root model it is a linear term
characterizedby the scalar parameter ρ which is the growthrate of
the root system. The diffusion correspondsto the spread of the root
biomass over space.It is described by the parameter σ which is ad ×
d matrix that characterizes the growth ofthe root biomass in any
direction exploiting freespace in the soil. The advection
corresponds to thedisplacement of the root biomass in a direction
andvelocity given by v which is a vector in Rd.
On the boundaries of Ω, what happens for thequantity being
transported is different dependingif the growth makes the roots to
come inside Ωor to go outside of Ω. If v is going inside Ω (atthe
inlet boundary) the root biomass u will enterthe domain and
increase. On edges where v isgoing out of the domain (outlet
boundary) the rootbiomass u is going to be pushed out of Ω.
Sincethis phenomena is oriented (causality) and thebehaviour of the
solution is different on inlet andoutlet boundaries, we need to
specify in the modelthese parts of the boundaries. Mathematically,
it isrequired to define the inlet boundary with respect
to v as
∂Ω− = {x ∈ ∂Ω : (v · n)(x) < 0} . (1)
The outlet boundary Ω+ is given by ∂Ω+ =∂Ω\∂Ω−. The dynamics of
the root system is stud-ied between the time t0 and t1 with 0 ≤ t0
< t1.The problem reads as follow: find u such that
∂tu+ v · ∇u−∇ · (σ∇u) + ρu = 0in ]t0, t1[×Ω
u(t0) = u0 at {t0} × Ωn · σ∇u = g on ]t0, t1[×∂Ω(n · v)u = gin
on ]t0, t1[×∂Ω−
(2)
where g ∈ L2(∂Ω) and gin ∈ L2(∂Ω−) are given.And u0 is the given
initial solution.
Problem (2) is known as the time dependentadvection-
reaction-diffusion problem and belongsto the class of parabolic
partial differential equa-tions. This equation is a model problem
that oftenoccurs in fluid mechanics but also in many
otherapplications in life sciences (see for instance [22],[23],
[24]).
Depending on the boundary conditions, theproblem has different
meanings. To simplifythe presentation we only consider the
Neumannboundary condition combined with an inlet bound-ary
condition at the inlet of the domain. The Neu-mann condition
specifies the value of the normalderivative of the solution at the
boundary of thedomain. The inlet boundary condition specifies
thequantity of u convected by v that enters in thedomain.
B. The weak problem
Since the goal is to solve the problem onunstructured meshes,
the spatial operators are ap-proximated using finite element
methods. Withinthis framework, it is classical to write the
problemin a variational form. Let us first introduce somefunctional
spaces [50].• The space H1(Ω) defined such that H1(Ω) ={v ∈ L2(Ω) :
∇v ∈ L2(Ω)} is a Hilbert spacewhen equipped with the norm ‖ · ‖1,Ω.
Werecall that ∀v ∈ H1(Ω), ‖v‖1,Ω = (v, v)1,Ω
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 3 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
and the scalar product (·, ·)1,Ω is defined by∀v ∈ H1(Ω),
(u, v)1,Ω =
∫Ωuv dx+
∫Ω∇u · ∇v dx.
• We denote L2(]t0, t1[, H) the space of H-valued functions
whose norm in H is inL2(]t0, t1[). This space is a Hilbert space
forthe norm
‖u‖L2(]t0,t1[,H) =(∫ t1
t0
‖u(t)‖2H)1/2
.
• Let B0 ⊂ B1 be two reflexive Hilbertspaces with continuous
embedding, we de-note W(B0, B1) the space of functions v :]t0,
t1[−→ B0 such that v ∈ L2(]t0, t1[, B0)and dtv ∈ L2(]t0, t1[, B1).
Equipped with thenorm
‖u‖W(B0,B1) = ‖u‖L2(]t0,t1[,B0)+‖dtu‖L2(]t0,t1[,B1),
the space W(B0, B1) is a Hilbert space [25].Using the previous
functional spaces, I now
define the problem in the following weak form:Find u in W such
that ∀v ∈ H
〈dtu(t),v(t)〉H′,H+a(t, u, v)=`(t,v) a.e. t∈]t0,t1[u(t0) =
u0,
(3)where W = W(H1(Ω), (H1(Ω))′) and H =H1(Ω) and
`(t, v) =
∫∂Ωg(t)vdγ (4)
a(t,u,v)=aA(t,u,v)+aD(t,u,v)+aR(t,u,v) (5)
with
aA(t, u, v) =
∫Ωv(v(t,x) · ∇u) dx, (6)
aD(t, u, v) =
∫Ω∇v · σ(t,x) · ∇u dx, (7)
aR(t, u, v) =
∫Ωρ(t)uv dx. (8)
One can prove that problems (3) and (2) areequivalent almost
everywhere in ]t0, t1[×Ω. Let us
assume that there is a constant σ0 > 0 such that
∀ξ ∈ Rd,d∑
i,j=1
σijξiξj ≥ σ0‖ξ‖2d a.e. in Ω. (9)
In addition, I assume that
infx∈Ω
(σ − 1
2(∇ · v)
)> 0 and inf
x∈∂Ω(v · n) ≥ 0.
(10)Under assumption (9) and (10), one can prove
that the problem is well-posed for sufficientlysmooth v, σ and ρ
(see for instance [25]).
C. The positivity preserving property of the solu-tion
In the framework of our applications to thesimulation of root
biomass densities one of thecrucial property of the problem is the
preservationof the positity of the solution along time. For
apositive initial solution u0, the solution of (3)
stayspositive.
Proposition II-C.1. Let u0 ∈ L2(Ω) and f ∈L2(]t0, t1[, L
2(Ω)). We consider u the solution of(3) in W . We assume that
u0(x) ≥ 0 a.e. in Ω andg(t,x) ≥ 0 a.e. in ]t0, t1[×∂Ω. Then u(t,x)
≥ 0a.e in ]t0, t1[×Ω.
Proof: I follow [25]. See also [26], [27]. Weconsider the
function u− defined by
u− =1
2(|u| − u).
Let us note that
u−=
{0 a.e in ]t0,t1[×Ω, if u≥0 a.e in ]t0,t1[×Ω,−u a.e in
]t0,t1[×Ω, if u
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
that are valid a.e in ]t0, t1[×Ω we can verify that
a(t, u−, u−) = −a(t, u, u−).
By adding the same quantity on both sides of theequation we
get
〈dtu−,u−〉+a(t,u−,u−)=〈dtu−,u−〉−a(t,u,u−).
Since u satisfy (3) we have
〈dtu−,u−〉+a(t,u−,u−) =〈dtu−,u−〉+〈dtu,u−〉− `(t, u−).
One can notice that 〈dtu−, u−〉 + 〈dtu, u−〉 = 0.Then we have
1
2dt‖u−‖20,Ω + a(t, u−, u−) = −`(t, u−) ≤ 0,
with g(t,x) ≥ 0 a.e. in ]t0, t1[×∂Ω. Now from thecoercivity of
the bilinear form a we obtain
1
2dt‖u−‖20,Ω + c‖u−‖20,Ω
≤ 12dt‖u−‖20,Ω + a(t;u−, u−) ≤ 0,
where c is a strictly positive constant. The estimateis then
1
2dt‖u−‖20,Ω ≤ −c‖u−‖20,Ω.
By the Gronwall lemma we have that
∀t ∈ [t0, t1]× Ω, ‖u−(t)‖20,Ω ≤ e−2ct‖u−(0)‖20,Ω.
Since c > 0 and t ≥ t0 ≥ 0 , we have that e−2ct ≤1 , so we
obtain
∀t ∈ [t0, t1]× Ω, ‖u−(t)‖20,Ω ≤ ‖u−(0)‖20,Ω.
Since u(0) = u0 ≥ 0 a.e in Ω we have u−(0) = 0a.e in Ω. So we
deduce that
∀t ∈ [t0, t1]× Ω, ‖u−(t)‖20,Ω ≤ 0.
But from the definition of u− we have u− ≥ 0 a.ein ]t0, t1[×Ω.
So we deduce that ‖u−(t)‖20,Ω = 0and thus u−(t) = 0 a.e in ]t0,
t1[×Ω. It means thatu ≥ 0 a.e in ]t0, t1[×Ω by definition of
u−.
III. APPROXIMATION OF THE MODEL
A. The operator splitting technique
Here we focus on the implementation ofthe operator splitting
technique. The time inter-val [t0, t1] is divided in N subspaces of
sizeδt such that [t0, t1] = ∪n=1,N ]tn, tn+1[ with∩n=1,N ]tn,
tn+1[= ∅. At each iteration step wesolve the following problems•
Find uA ∈ H such that ∀v ∈ H , for a.e t ∈
]tn, tn+1[,
〈dtuA(t), v(t)〉H′,H + aA(t, uA, v) = 0uA(tn) = u(tn).
• Find uD ∈ H such that ∀v ∈ H , for a.et ∈]tn, tn+1[,
〈dtuD(t), v(t)〉H′,H + aD(t, uD, v) = `(t, v)uD(tn) =
uA(tn+1).
• Find uR ∈ H such that ∀v ∈ H , for a.e t ∈]tn, tn+1[,
〈dtuR(t), v(t)〉H′,H + aR(t, uR, v) = 0uR(tn) = uD(tn+1).
• Set u(tn+1) = uR(tn+1).The bilinear forms aA(t, u, v), aD(t,
u, v) andaR(t, u, v) are respectively given by (6), (7) and(8). And
`(t, v) is the linear form (4). If theoperators are commutative,
then the splitting errorvanishes. Otherwise, if the operators are
not com-mutative, then the splitting error does not vanishand a
second order splitting would be required(see [6]). In the
following, I present the differentschemes related to each
operator.
B. The advection step: DG upwind scheme
The advection step consists in solving the fol-lowing transport
problem : Find u such that ∀v ∈H , for a.e t ∈]tn, tn+1[,
〈dtu(t), v(t)〉H′,H + aA(t, u, v) = 0 (12)uA(tn) = uR(tn)
(13)
where aA(t, u, v) is the bilinear form (6). For thespace
approximation of this problem, we imple-mented the DG upwind method
presented below.
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 5 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
Let Th be a regular family of decomposition intriangles of the
domain Ω such that
Ω =
N⋃i=1
K̄i and Ki ∩Kj = ∅,∀i 6= j.
The h subscript in Th denotes the size of the meshcells and it
is defined by
h = maxK∈Th
hK
where hK is the diameter of the element K. Let Ehbe the set of
edges of the elements of Th. Amongthe elements of Eh we denote by
Ebh the set ofedges belonging to ∂Ω. The sets Eb,−h and E
b,+h
are the sets of edges belonging to ∂Ω− and ∂Ω+
respectively. And E ih is the set of interior edges.Let us
consider an element of E ih. We denote byT+ and T− the two mesh
elements sharing theedge e so that e = ∂T+ ∩ ∂T− where the minusand
plus superscripts depend on the direction ofthe advection vector.
By convention we supposethat v goes from T− to T+ that is v ·n+e
< 0 andv · n−e > 0 where n+e (resp. n−e ) is the
outwardnormal vector of e in T+ (resp. T−). When it isnot necessary
to distinguish the orientation of thenormal vectors n+e and n
−e we denote by n the
unitary normal of e.Let us consider the advection problem on
each
element Ki of the domain : for all Ki, i = 1, Nwe look for u the
solution of the equation (12)defined on Ki. Similarly to the
problem definedon all the domain Ω, we look for a solution u thatis
in L2(Ki) and such that ∇u is in L2(Ki) for allKi in Th. Let us
introduce the following brokenSobolev space:
H1(Th) ={v ∈ L2(Ω) : ∇v ∈ L2(Ki)
and v ∈ H1/2+ε(Ki), ∀Ki ∈ Th}
with ε a positive real number. The trace of thefunctions of
H1(Th) are meaningful on e ⊂ Ki,∀Ki ∈ Th. The functions v of H1(Th)
have twotraces along the edges e. We denote v+e the traceof v along
e on the side of triangle T+ and v−e thetrace of v along e on the
side of T−. On edges
that are subsets of ∂Ω the trace is unique and wecan note
v+e = v if e ∈ Eb,−h and v
−e = v if e ∈ E
b,+h ,
and by convention, we set
v−e = 0 if e ∈ Eb,−h and v
+e = 0 if e ∈ E
b,+h .
The jump of functions of H1(Th) across the inter-nal edge e is
defined by:
JvK = v+e − v−e , ∀e ∈ E ih.For edges belonging to the boundary
of Ω we take
JvK = ve,∀e ∈ Eb,−h and JvK = −ve, ∀e ∈ Eb,+h ,
with ve the trace of v along e. The mean value ofu on e is
defined by
{{v}} = 12
(v+e + v−e ),∀e ∈ E ih.
Besides for edges on the boundaries we take
{{v}} = ve, ∀e ∈ Ebh.Let us denote by X the functional space
definedsuch that
X = {v : ]t0, t1[−→ H1(Th) :v ∈ L2(]t0, t1[, H1(Th));and dtv ∈
L2(]t0, t1[, H1(Th)′)}.
This space is a Hilbert space equipped with thenorm
‖v‖X = ‖v‖L2(]t0,t1[,H1(Th)) +‖v‖L2(]t0,t1[,H1(Th)′).The DG
variational formulation of the advection
step written on the broken Sobolev space takesthe following
form: Find u in X such that for a.et ∈]t0, t1[, ∀v ∈ H1(Th)〈dtu(t),
v(t)〉H1(Th)′,H1(Th)+a
uph (t;u,v)=`
uph (t;v),
u(t0) = u0,
where the form auph (t;u, v) is the approximationof the
advection term. It consists in the upwindformulation of the DG
method [28]. It reads:
auph (t;u, v)=∑K∈Th
∫Ku(ρv−v · ∇v)dx
−∑
e∈Eb,±h ∪Eb,+h
∫e|v · n+e |u−e JvKds. (14)
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 6 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
The approximated linear form of the the right handside reads
`uph (t; v) = −∑e∈Eb,−h
∫e(v · n+e )ginv+e ds.
The DG-formulation (14) is consistent and stable,see for example
[32]. The discontinuous Galerkinmethod consists in searching the
solution in theapproximation space Xh defined such that
Xh ={v :]t0, t1[−→W kh ; v ∈ L2(]t0, t1[,W kh );
and dtv ∈ L2(]t0, t1[, (W kh )′)},
where W kh is given by
W kh ={vh ∈ L2(Ω);∀K ∈ Th, vh|K ∈ Pk
}.
Let us note that the functions of W kh can bediscontinuous from
one element of the mesh to theother. Let us note that W kh is
embedded in H
1(Th)so that Xh ⊂ X . This problem can be writtenin a matrix
form. Let us denote (λi)i=1,n thebasis of the finite dimensional
subspace W kh wheren = dim(W kh ). In this basis the
approximatedsolution takes the form:
uh(t, x, y) =
n∑i=1
ξi(t)λi(x, y),
where the ξi(t) are the degrees of freedom. Let usdefine X the
vector of degrees of freedom:
X(t) = (ξ1(t), . . . , ξn(t))T .
The approximated problem then reduces to findX(t) ∈ [C2(0, T )]n
such that
MdX(t)
dt+ Aup(t)X(t) = Luph (t)
MX(0) = MX0
where M and Aup(t) are two matrices definedsuch that
M = (Mi,j)i,j and Mi,j =∑T∈Th
∫Kλiλjdx,
(15)
Aup =(Aupi,j
)i,j
and Aupi,j = auph (t; , u, v), (16)
and Luph (t) is the vector of size n defined such that(Luph
(t)
)i
= `uph (t;λi) for i = 1, n. The problemreduces to a linear
system of ordinary differentialequations. The time approximation is
based on afinite difference scheme.
At each iteration step we solve the followingproblem: Find XN+1
∈ Rn such that
1
δtM(XN+1 −XN
)+ (1− θ)AupXN + θAupXN+1 (17)= (1− θ)Lup,Nh + θL
up,N+1h
and MX0 = MX0,
where θ is a real parameter taken in [0, 1]. Forθ = 0, we have
the explicit Euler schema. Forθ = 1, it is the implicit Euler
schema. For θ = 1/2,it is the Crank-Nicolson schema.
C. The diffusion step
The diffusion step consists in solving the fol-lowing problem :
Find u such that ∀v ∈ H , fora.e t ∈]tn, tn+1[,
〈dtu(t), v(t)〉H′,H + aD(t;u, v) = `(t; v)u(tn) = uA(tn)
where aD(t;u, v) is the bilinear form (7) and`(t; v) is the
linear form (4). In the setting intro-duced before, the DG
variational formulation ofthe diffusion step written in the broken
Sobolevspace takes the following form: Find u in X suchthat ∀v ∈
H1(Th), for a.e. t ∈]t0, t1[
〈dtu(t), v〉H1(Th)′,H1(Th) + aiph (t;u, v) = `
iph (t; v)
u(t0) = u0.
The form aiph (t;u, v) is the approximation of thediffusion
term. It consists in the interior penalty
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 7 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
formulation (IP) that reads
aiph (t;u, v) =∑K∈Th
∫Kσ∇u · ∇v dx
−∑e∈Eih
∫e{{σ∇u}} · n+e JvK ds
+∑e∈Eih
∫e{{σ∇v}} · n+e JuK ds
+∑e∈Eih
η
he
∫eJuKJvK ds,
where η is a positive penalization factor. The linear
form `iph (t; v) is given by `iph (t; v) =
∑e∈Ebh
∫egv ds.
This formulation was introduced in [31] andis known as the
non-symmetric interior penalty(NSIP) formulation, see [30], [32].
In matrix formthe problem reduces to find X(t) ∈ [C2(0, T )]nsuch
that
MdX(t)
dt+ Aip(t)X(t) = Liph (t)
MX(0) = MX0
where M is defined by (15) and Aip is definedsuch that
Aip =(Aipi,j
)i,j
and Aupi,j = aiph (t; , u, v).
The vector Liph (t) is such that(Liph (t)
)i
=
`iph (t;λi) for i = 1, n. Similarly to the advectionstep, the
time approximation of the problem isbased on a finite difference
scheme of the form(17).
D. The reaction step
The reaction step consists in solving the follow-ing problem :
Find u such that ∀v ∈ H , for a.e.t ∈]tn, tn+1[
〈dtu(t), v(t)〉H′,H + aR(t;u, v) = 0u(tn) = uD(tn)
where aR(t;u, v) is the bilinear form (8). Thisproblem takes the
following matrix form find
X(t) ∈ [C2(0, T )]n such that
dX(t)
dt+ ρX(t) = 0
X(0) = X0
where we recall that ρ is a constant real parameter.This problem
can be solved by an exact scheme (akind of schemes that provide
exact solutions, i.e. asolution equal to the analytical solution).
At eachiteration we find XN+1 such that
1
Φ(δt)
(XN+1 −XN
)= −ρXN
with Φ(δt) = 1ρ(1 − exp(−ρδt)). This schemeis unconditionally
stable, meaning that we canchoose the time step independently from
the spacestep. It is also positively stable, meaning that ifXN ≥ 0
so is XN+1.
IV. VALIDATION OF THE SPLITTINGALGORITHM WITH A SIMPLE TEST
CASE
Problem (3) has been already solved using dis-continuous
Galerkin elements (DG) [3]. Advectionand Diffusion operators were
solved simultane-ously using the Crank-Nicolson scheme
providingstable results. However, even for simple test casessome
simulations did not always provide positivenumerical solutions. One
reason is that the sametime approximation scheme is not necessarily
suit-able for both the advection and for the diffusion.That is why
a new operator splitting algorithm hasbeen implemented with a
different time scheme foreach operator.
The goal of this section is to validate the im-plementation of
the code. To this end I comparethe convergence of the approximation
with andwithout the splitting technique. I briefly explorethe
question of the positivity of the approximatedsolution.
A. Description of the simple test-case
First let me introduce a simplified test-case forthe validation
of the splitting algorithm. Set L > 0,and Ω =] − L;L[2. Let v =
(v1, v2) ∈ R2 and
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 8 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
d ∈ R be a constant and 0 ≤ t0 ≤ t1. Find u suchthat
∂u
∂t+ v · ∇u+ ρu = d∆u in ]t0, t1[×Ω,
u(x, y, 0) = u0(x, y) on {t0} × Ω, (18)n · ∇u = g on ]t0,
t1[×∂Ω.n · vu = gin on ]t0, t1[×∂Ω−.
The initial condition and the boundary conditionare chosen such
that the solution of problem (18)is explicitly given by ∀(x, y, t)
∈ Ω×]t0, t1[
u(x, y, t) = c0
(a2
a2 + td
)κ(x, y, t)e−ρt.
with
κ(x, y, t)
= c0 exp
(−(x− x0 − tv1)
2 + (y − y0 − tv2)2
4(a2 + td)
)where c0 > 0, a > 0, x0 and y0 are real parametersand v1
and v2 are the two components of v. Noticethat u(x, y, t) > 0
for all (x, y, t) in Ω×]t0, t1[.
B. Numerical validation and convergence
To validate the implementation of the splittingtechnique, I ran
the previous test case with differ-ent mesh sizes and time steps
and I computed theglobal L2-errors such that
eh =
(δt
N∑k=1
‖u(tk)− uh(tk)‖20,Ω
)1/2where tk = t0 + kδt, with k ∈ N+∗ and tN = t1.
The flexibility of the splitting technique allowsto choose
different time schemes for each operator.I consider a θ-scheme with
θ = 0 (explicit Euler),θ = 1 (implicit Euler), and θ = 12
(Crank-Nicolson) for both the advection step and thediffusion step,
and I consider an exact scheme forthe reaction step. For the
simulations I took theparameters such that v = (0.1, 0)T , σ = 0.01
andρ = −1. The triangular meshes used for the simu-lations are
identified by h which is the size of thebiggest triangle of the
mesh. Table I, page 9, givesthe number of triangles and the number
of nodesof each mesh used for the simulations. Choosing
h (≈) number of triangles number of nodes2.63× 10−1 68 451.31×
10−1 272 1576.57× 10−2 1 088 5853.29× 10−2 4 352 2 2571.64× 10−2 17
408 8 8658.22× 10−3 69 632 35 1374.11× 10−3 278 528 139 905
TABLE ITRIANGULAR MESHES USED FOR THE SIMULATIONS.
Fig. 1. Solution of the validation test case at t = t0 (left)and
t = t1 (right) computed using the DG method with p1-finite elements
and the Euler implicit scheme (θ = 1) andthe operator splitting
technique with h ≈ 8.2 × 10−3 andδt = 10−2.
L = 1/2, the simulations are performed betweent0 = 0 and t1 = 1
for different values of the timestep δt. Fig. 1, page 9, shows the
solution at t = t0and t = t1. The code is implemented in Fortran90
and it is run under a 64-bit Linux operatingsystem on a 8-core
processor Intel R©CoreTMi7-7820HQ at a frequency of 2.9GHz and with
32 GBof RAM. The sparse matrices resulting from thefinite element
approximation are inverted using asolver provided by the library
MUMPS [51], [52].
According to Fig. 2, page 10, all the three tem-poral schemes
provide results with approximatelythe same level of accuracy with a
spatial con-vergence rate of 2 computed with the global L2-
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 9 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
Fig. 2. Convergence of the solution with respect to the
meshsize: plot of the total L2-error computed between t = 0 andt =
1 with and without the splitting technique for the explicitEuler
scheme (θ = 0), the implicit Euler scheme (θ = 1) andthe
Crank-Nicolson scheme (θ = 1/2) for δt = 5× 10−5.
Fig. 3. Convergence of the solution with respect to the
timestep: plot of the total L2-error computed between t = 0 andt =
1 with and without the splitting technique for the implicitEuler
scheme (θ = 1) and the Crank-Nicolson scheme (θ =1/2) for h = 4.1×
10−3.
norm. The same order of convergence is obtainedwhen the problem
is solved without the splittingtechnique.
Figure 3 on page 10 shows that the Crank-Nicolson scheme (θ =
1/2) converges in δt2 whilethe Euler Implicit scheme (θ = 1)
converges inδt, with and without the splitting technique.
Theconvergence rate in time has to be computed witha really refined
mesh grid (here h ≈ 4.1× 10−3).
Fig. 4. Validation of the test case: plot of the CPU timeagainst
the mesh size (h) for the computations performed witha processor
Intel R©CoreTMi7-7820HQ at 2.9 GHz and RAM32 GB, between t = 0 and
t = 1 with and without thesplitting technique for the explicit
Euler scheme (θ = 0),the implicit Euler scheme (θ = 1) and the
Crank-Nicolsonscheme (θ = 1/2) for δt = 5× 10−5.
Fig. 5. Validation of the test case: plot of the CPU timeagainst
the time step (δt) for the computations performed witha processor
Intel R©CoreTMi7-7820HQ at 2.9 GHz and RAM32 GB, between t = 0 and
t = 1 with and without thesplitting technique for the explicit
Euler scheme (θ = 0),the implicit Euler scheme (θ = 1) and the
Crank-Nicolsonscheme (θ = 1/2) for h ≈ 4.1 × 10−3 and δt ranging
from5× 10−1 to 5× 10−4. Note that the computations performedhere
with θ = 0 gave unstable results.
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 10 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
δt global L2-errors min dof t+ CPU time1 · 10−3 unstable
unstable - 95 s2 · 10−4 1.24 · 10−4 −1 · 10−4 0.221 475 s1 · 10−4
1.22 · 10−4 −1 · 10−4 0.218 838 s5 · 10−5 1.21 · 10−4 −1 · 10−4
0.216 1672 s2.5 · 10−5 1.20 · 10−4 −9 · 10−5 0.215 3376 s1 · 10−5
1.20 · 10−4 −9 · 10−5 0.215 10183 s
TABLE IICOMPUTATIONS PERFORMED WITH THE SPLITTING TECHNIQUE AND
THE EXPLICIT EULER SCHEME (θ = 0) WITH
h ≈ 1.6 · 10−2 (INTEL R©CORETM I7-7820HQ AT 2.9 GHZ, RAM 32
GB).
It results an additional cost in term of CPU time,since it
behaves like 1/h2, as shown on figure 4page 10. For bigger values
of h the plot of theerrors gave convergence order in time less
than1 and 2 for the implicit Euler scheme and theCrank-Nicolson
scheme respectively. As expected,the explicit Euler scheme is
conditionally stable,such that, when the CFL condition is
fulfilled, thecomputational time becomes prohibitive. Indeed,it
behaves like 1/δt, as shown on figure 5. Forinstance, the
computation with h ≈ 4.1 × 10−3and δt = 10−5 takes more than 30
hours with thedevice specified above. That is why, in the rest
ofthe paper, we will only focus on implicit Euler andCrank-Nicolson
schemes. However I present hereadditional computations performed
with a biggermesh size (h ≈ 1.6×10−2) and smaller time stepschosen
such that the CFL condition is fulfilled.The global L2-errors and
the CPU time are shownon table II. Clearly, the mesh is not fine
enoughto recover the convergence order in δt, indeeddecreasing the
time step results only in an increaseof the computational time but
not in a significantdecrease of the errors.
C. Some comments on the positivity
1) Positivity of the full problem: Table III onpage 11 and table
V on page 12 give the min-imum values of the degrees of freedom
(dof)obtained during the simulations performed respec-tively with
and without the splitting technique.The minimum value of the dof is
defined suchthat mintk(mini=1,nXki ) where X
ki is the i
th dofat time tk. This quantity gives an idea about the
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 −7 · 10−1 −7 · 10−1
−7 · 10−12 · 10−1 −2 · 10−1 −2 · 10−1 −2 · 10−11 · 10−1 −8 · 10−4
−1 · 10−3 −2 · 10−34 · 10−2 −2 · 10−4 −6 · 10−4 −1 · 10−32 · 10−2
−3 · 10−13 −2 · 10−4 −6 · 10−41 · 10−2 −6 · 10−5 −2 · 10−20 −1 ·
10−44 · 10−3 −3 · 10−4 4 · 10−65 5 · 10−662 · 10−3 −3 · 10−4 −9 ·
10−11 5 · 10−881 · 10−3 −2 · 10−4 −8 · 10−10 8 · 10−1141 · 10−4 −9
· 10−5 −1 · 10−11 −4 · 10−35
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 −5 · 10−8 −9 · 10−9
−2 · 10−92 · 10−1 4 · 10−9 1 · 10−9 4 · 10−101 · 10−1 3 · 10−11 3 ·
10−11 1 · 10−114 · 10−2 6 · 10−17 5 · 10−17 5 · 10−172 · 10−2 3 ·
10−23 2 · 10−23 2 · 10−231 · 10−2 1 · 10−31 4 · 10−32 3 · 10−324 ·
10−3 −3 · 10−5 1 · 10−48 6 · 10−492 · 10−3 −5 · 10−5 2 · 10−65 2 ·
10−661 · 10−3 −7 · 10−5 −1 · 10−13 2 · 10−881 · 10−4 −9 · 10−5 −7 ·
10−12 −3 · 10−43
Implicit Euler scheme (θ = 1)
TABLE IIIMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED
WITH THE SPLITTING ALGORITHM WITH THECRANK-NICOLSON SCHEME (TOP)
AND THE IMPLICIT
EULER SCHEME (BOTTOM).
stability and the positivity preserving behaviour ofthe schemes.
Tables III and V clearly show thatthe schemes are not always
positivity preserving.In case where the approximated solution is
notpositive for all t > t0 I also check if it
becomesnon-negative for larger time ie. if there is t+ > t0
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 11 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
δt h≈1.6·10−2 h≈8.2·10−3 h ≈4.1·10−35 · 10−1 - - -2 · 10−1 - -
-1 · 10−1 - - -4 · 10−2 - - -2 · 10−2 0.24 - -1 · 10−2 0.07 0.10 -4
· 10−3 0.172 t0 t02 · 10−3 0.202 0.044 t01 · 10−3 0.210 0.079 t01 ·
10−4 0.2140 0.0939 0.0297
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 1 1 12 · 10−1 t0 t0
t01 · 10−1 t0 t0 t04 · 10−2 t0 t0 t02 · 10−2 t0 t0 t01 · 10−2 t0 t0
t04 · 10−3 0.072 t0 t02 · 10−3 0.132 t0 t01 · 10−3 0.173 0.029 t01
· 10−4 0.2102 0.0874 0.0217
Implicit Euler scheme (θ = 1)
TABLE IVPOSITIVITY THRESHOLD (t+) COMPUTED WITH THE
SPLITTING ALGORITHM AND THE CRANK-NICOLSONSCHEME (TOP) AND THE
IMPLICIT EULER SCHEME
(BOTTOM).
such that Xki ≥ 0, ∀i = 1, n for all tk > t+ > t0.The
smallest such t+, if it exists, is referred asthe positivity
threshold, as defined in [36]. TableIV on page 12 and table VI on
page 13 give thepositivity thresholds computed with and withoutthe
splitting technique respectively.
For the Crank-Nicolson scheme (θ = 1/2) andthe implicit Euler
scheme (θ = 1) the positivityis obtained under a specific condition
on the timestep and the mesh size. For a given mesh size,the time
step δt must be bounded from above,but also from below to guarantee
that the solutionstays positive all along the simulation. In the
caseof the splitting technique those bounds are morerestrictive
than in the case of the resolution ofthe full problem without
splitting. Those boundsare also more restrictive in the case of the
Crank-Nicolson (θ = 1/2) scheme than in the case of the
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 −4 · 10−1 −4 · 10−1
−4 · 10−12 · 10−1 −1 · 10−1 −1 · 10−1 −1 · 10−11 · 10−1 1 · 10−13 1
· 10−13 1 · 10−134 · 10−2 1 · 10−21 1 · 10−21 9 · 10−222 · 10−2 3 ·
10−30 1 · 10−30 1 · 10−301 · 10−2 −6 · 10−5 1 · 10−42 9 · 10−434 ·
10−3 −3 · 10−4 3 · 10−64 5 · 10−652 · 10−3 −3 · 10−4 −8 · 10−11 3 ·
10−871 · 10−3 −2 · 10−4 −7 · 10−10 3 · 10−1131 · 10−4 −9 · 10−5 −1
· 10−11 −8 · 10−35
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 2 · 10−5 2 · 10−5 2
· 10−52 · 10−1 1 · 10−7 1 · 10−7 1 · 10−71 · 10−1 6 · 10−10 5 ·
10−10 5 · 10−104 · 10−2 1 · 10−15 1 · 10−15 1 · 10−152 · 10−2 6 ·
10−22 5 · 10−22 5 · 10−221 · 10−2 1 · 10−30 7 · 10−31 6 · 10−314 ·
10−3 −2 · 10−5 2 · 10−47 8 · 10−482 · 10−3 −4 · 10−5 2 · 10−64 2 ·
10−651 · 10−3 −7 · 10−5 −3 · 10−13 2 · 10−871 · 10−4 −9 · 10−5 −7 ·
10−12 −1 · 10−42
Implicit Euler scheme (θ = 1)
TABLE VMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED
WITHOUT THE SPLITTING ALGORITHM AND THECRANK-NICOLSON SCHEME
(TOP) AND IMPLICIT EULER
SCHEME (BOTTOM).
implicit Euler scheme (θ = 1). Refining the meshresults in less
restrictions on the time step but alsolead to additional
computational time.
With the Crank-Nicolson scheme (θ = 1/2),for a given mesh size,
if δt is too big, there isno threshold of positivity in tk ∈]t0,
t1] and thecomputed solution is not non-negative all alongthe
simulation. For θ = 1/2 and θ = 1, stillwith a given mesh size, if
δt is too small, thesimulations showed that there is a threshold
ofpositivity t+ such that the approximated solutionbecomes
non-negative for tk ≥ t+. The thresholdsof positivity slightly
depend on the time stepand tend to increase when the time step δt
isdecreased. The computations clearly showed thatthe positivity
thresholds diminish with the meshsize h (see for example [36]).
Altogether, the positivity of the approximated
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 12 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 - 1 12 · 10−1 0.8
0.8 0.81 · 10−1 t0 t0 t04 · 10−2 t0 t0 t02 · 10−2 t0 t0 t01 · 10−2
0.08 t0 t04 · 10−3 0.188 t0 t02 · 10−3 0.208 0.050 t01 · 10−3 0.213
0.083 t01 · 10−4 0.2143 0.0942 0.0300
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 t0 t0 t02 · 10−1 t0
t0 t01 · 10−1 t0 t0 t04 · 10−2 t0 t0 t02 · 10−2 t0 t0 t01 · 10−2 t0
t0 t04 · 10−3 0.0720 t0 t02 · 10−3 0.1380 t0 t01 · 10−3 0.1760
0.0290 t01 · 10−4 0.2104 0.0877 0.0221
Implicit Euler scheme (θ = 1)
TABLE VIPOSITIVITY THRESHOLD (t+) COMPUTED WITHOUT THE
SPLITTING ALGORITHM AND THE CRANK-NICOLSONSCHEME (TOP) AND THE
IMPLICIT EULER SCHEME
(BOTTOM).
solution is obtained at the expense of the compu-tational cost,
but for a given mesh size h computa-tions performed with too small
time step can alsolead to a loss of positivity for small tk. In
[36] (andreferences therein), Thomée showed that thresholdvalues
of tk > 0 may exist such that X(t) > 0when t > tk.
At this stage, one may wonder how each term ofthe splitting
behaves in terms of positivity preser-vation. The reaction term is
approximated usingan exact scheme, so obviously the positivity of
thesolution is preserved. What about the diffusion andthe advection
term ?
2) Positivity of the pure diffusion problem:Here I set v = (0,
0) and ρ = 0, while keeping allothers parameters to the same values
as previously.Table VII clearly shows that the Crank-Nicolsonscheme
(θ = 1/2) is positivity preserving under a
δt h≈1.6·10−2 h≈8.2·10−3 h ≈4.1·10−35 · 10−1 −5 · 10−1 −5 · 10−1
−5 · 10−12 · 10−1 −2 · 10−1 −2 · 10−1 −2 · 10−11 · 10−1 4 · 10−15 4
· 10−15 4 · 10−154 · 10−2 6 · 10−23 5 · 10−23 4 · 10−232 · 10−2 2 ·
10−31 8 · 10−32 6 · 10321 · 10−2 −4 · 10−5 1 · 10−43 6 · 10434 ·
10−3 −3 · 10−4 4 · 10−65 5 · 10−662 · 10−3 −3 · 10−4 −5 · 10−11 5 ·
10−881 · 10−3 −2 · 10−4 −6 · 10−10 8 · 10−1141 · 10−4 −9 · 10−5 −1
· 10−11 −1 · 10−34
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 4 · 10−6 4 · 10−6 4
· 10−52 · 10−1 2 · 10−8 2 · 10−8 2 · 10−81 · 10−1 2 · 10−11 2 ·
10−11 2 · 10−114 · 10−2 5 · 10−17 5 · 10−17 5 · 10−172 · 10−2 3 ·
10−23 2 · 10−23 2 · 10−231 · 10−2 1 · 10−31 4 · 10−32 3 · 10−324 ·
10−3 −2 · 10−5 1 · 10−48 6 · 10−492 · 10−3 −4 · 10−5 2 · 10−65 2 ·
10−661 · 10−3 −7 · 10−5 −3 · 10−13 2 · 10−881 · 10−4 −9 · 10−5 −7 ·
10−12 −2 · 10−42
Implicit Euler scheme (θ = 1)
TABLE VIIMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED
FOR THE PURE DIFFUSION PROBLEM WITH THECRANK-NICOLSON SCHEME
(TOP) AND THE IMPLICIT
EULER SCHEME (BOTTOM).
CFL-like condition with upper and lower bounds,like in the
previous test. The implicit Euler scheme(θ = 1) seems to be more
favorable, since itpreserves the positivity even for big values of
thetime step. For both the Crank-Nicolson (θ = 1/2)and implicit
Euler (θ = 1) schemes, the approx-imated solution suffers from a
loss of positivityfor small values of tk when the time step is
toosmall. According to table VIII, there are positivitythresholds,
like in [36] which indeed deals withthe heat equation.
3) Positivity of the pure advection problem:Here I set σ = 0 and
ρ = 0, while keeping allothers parameters to the same values as in
the firsttest. Table IX shows that none of the
computationsperformed gave a non negative solutions, eventhough the
minimum value of the dof can be reallyclose to zero for small mesh
sizes. Besides, I did
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 13 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 - - -2 · 10−1 0.8
0.8 0.81 · 10−1 t0 t0 t04 · 10−2 t0 t0 t02 · 10−2 t0 t0 t01 · 10−2
0.1 t0 t04 · 10−3 0.2 t0 t02 · 10−3 0.216 0.054 t01 · 10−3 0.219
0.085 t01 · 10−4 0.2203 0.0956 0.0303
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6· 10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 t0 t0 t02 · 10−1 t0
t0 t01 · 10−1 t0 t0 t04 · 10−2 t0 t0 t02 · 10−2 t0 t0 t01 · 10−2 t0
t0 t04 · 10−3 0.084 t0 t02 · 10−3 0.148 t0 t01 · 10−3 0.184 0.032
t01 · 10−4 0.2165 0.0893 0.0224
Implicit Euler scheme (θ = 1)
TABLE VIIIPOSITIVITY THRESHOLD (t+) COMPUTED FOR THE PURE
DIFFUSION PROBLEM WITH THE CRANK-NICOLSONSCHEME (TOP) AND THE
IMPLICIT EULER SCHEME
(BOTTOM).
not observe any positivity threshold. The approx-imated solution
stays non positive all along thesimulation. However I run
additional simulationswith even smaller mesh size (h ≈ 2.0 ×
10−3and δt = 10−4). This time the computed solutionwas positive at
the beginning of the simulation(before t− = 1.9 × 10−3), pointing
the existenceof a threshold of negativity, to finally reaching
anegative minimum values of dof (around −10−44).Unfortunately, this
threshold of negativity is reallysmall compared to the ending time
of the compu-tation (t1 = 1), while the computational time
wasreaching more than 14 hours (Intel R©CoreTMi7-7820HQ at 2.9 GHz,
RAM 32 GB) for both theCrank-Nicolson and the implicit Euler
schemes.
In fact it is well known that for the advectionterm the solution
can be polluted by overshoot andundershoot oscillations near a
discontinuity or a
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 −2 · 10−1 −2 · 10−1
−2 · 10−12 · 10−1 −4 · 10−2 −4 · 10−2 −4 · 10−21 · 10−1 −4 · 10−3
−2 · 10−3 −1 · 10−34 · 10−2 −1 · 10−3 −4 · 10−7 −5 · 10−72 · 10−2
−1 · 10−3 −6 · 10−8 −2 · 10−131 · 10−2 −1 · 10−3 −4 · 10−8 −3 ·
10−284 · 10−3 −1 · 10−3 −3 · 10−8 −1 · 10−282 · 10−3 −1 · 10−3 −3 ·
10−8 −1 · 10−281 · 10−3 −1 · 10−3 −3 · 10−8 −1 · 10−281 · 10−4 −1 ·
10−3 −3 · 10−8 −1 · 10−28
Crank-Nicolson scheme (θ = 1/2)
δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−35 · 10−1 −1 · 10−4 −1 · 10−10
−8 · 10−342 · 10−1 −2 · 10−4 −6 · 10−10 −3 · 10−321 · 10−1 −4 ·
10−4 −1 · 10−9 −2 · 10−314 · 10−2 −6 · 10−4 −4 · 10−9 −9 · 10−312 ·
10−2 −8 · 10−4 −7 · 10−9 −2 · 10−301 · 10−2 −9 · 10−4 −1 · 10−8 −5
· 10−304 · 10−3 −1 · 10−3 −1 · 10−8 −8 · 10−302 · 10−3 −1 · 10−3 −2
· 10−8 −2 · 10−291 · 10−3 −1 · 10−3 −2 · 10−8 −4 · 10−291 · 10−4 −1
· 10−3 −3 · 10−8 −9 · 10−29
Implicit Euler scheme (θ = 1)
TABLE IXMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED
FOR THE PURE ADVECTION PROBLEM WITH THECRANK-NICOLSON SCHEME
(TOP) AND THE IMPLICIT
EULER SCHEME (BOTTOM).
sharp layer, see [34], [33], [35], [30]. For low orderaccurate
spacial approximations one can prove thepositivity preserving
property of the scheme [33].But for high order schemes slopes
limiters areoften required to guarantee the positivity of
theapproximated solution. When slope limiters areused, explicit
time schemes seem to be suitablefor the advection [6]. However, in
the next sectionwe will only privilege a numerical scheme thatis
unconditionally stable, i.e. the Crank-Nicolsonscheme, that is a
two-order scheme.
V. APPLICATION TO THE SIMULATION OF ROOTSYSTEM GROWTH
In this section, I apply the previous DG-splittingapproach to
solve numerically the C-Root model.First, I detail the parameters
used for the simula-tions, then, I present and validate the results
of the
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 14 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
simulations.
A. The C-Root parameters for Eucalyptus rootgrowth
The parameters and operators’ coefficients arechosen based on
the previous calibration done in[2]. The diffusion coefficient, σ,
is build using thefollowing Gaussian function
fα,µ(x, y) =α√2π
exp
(−(r(x, y)− µ)
2
2
)
where r(x, y) =√
(x− x0)2 + (y − y0)2 and(x0, y0) ∈ Ω =] − L,L[. The function
fα,µ(x, y)depends on two real and positive parameters: α,related to
the maximum amplitude of fα,µ, and µ,the distance from (x0, y0) to
the point where thefunction fα,µ reaches its maximum.
The diffusion tensor is taken such that
σ(x, y) = fαd,µd(x, y)
(1 00 1
),
for all (x, y) ∈ Ω, and αd, µd ∈ R+ are givenparameters. The
advection vector is taken such thatv(x, y) = (0,−v0)T , for all (x,
y) ∈ Ω, with v0a positive constant. The reaction term is constantin
space and splited into two contributions: βr andµr, the branching
and mortality rates, respectively.That is
ρ = βr − µr ∈ R.
The branching rate, βr, is estimated from biologi-cal knowledge:
it is equal to zero before 9 monthsand equal to 1/3 after, since no
roots die before 9month. However, for the following simulations
wewill not distinguish the contribution of βr and µr,so that the
reaction term will only be described bythe parameter ρ.
Fig. 6. Density of apices computed at t = 6, t = 12, t = 18and t
= 24 months (from the left to the right and from thetop to the
bottom).
B. Some simulations
For the simulation the initial solution is chosenequal to the
following function:
u0(x, y) = A
[exp(b(1− x))
(exp(−b(1− x)) + exp(b(1− x)))
− exp(b(−1− x))(exp(−b(−1− x)) + exp(b(−1− x)))
]×[
exp(b(1− y))(exp(−b(1− y)) + exp(b(1− y)))
− exp(b(−1− y))(exp(−b(−1− y)) + exp(b(−1− y)))
]with A = 2 · 10−4 and b = 1. The parameters’values µr, αd , µd
are estimated using the code
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 15 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
Fig. 7. L2-error with respect to the solution obtained withthe
mesh of size h ≈ 9.87× 10−2 and δt = 10−3 computedat t = 6, t = 12,
t = 18 and t = 24 months and plottedagainst the mesh size.
described in [2]. I run the simulations from t0 = 1to t1 = 24
months, with L = 13. The simulationsare performed for different
values of the mesh size.Fig. 6, page 15, shows the solution
computed atfour different stages of the root system develop-ment.
One can notice the diffusion of the apicesin the soil and also the
transport of the apicesfrom the top to the bottom of the soil
layer. Sincethere is no analytic solution, the convergence ofthe
computation is evaluated by measuring the L2-errors with respect to
the approximated solutioncomputed with the finest mesh (h ≈ 8.97×
10−2)and with δt = 10−3. The curves of the errorsagainst the mesh
size are plotted on figure 7and clearly show that the DG-splitting
algorithmconverges with a convergence rate of almost two.However,
one can note that the mesh sizes and thetime steps chosen for the
simulations presentedhere might not be small enough. The
positivityof the solution is not preserved at all times andthe full
convergence might not be acheived. Un-fortunatly, refining the mesh
sizes and the timesteps can lead to prohibitive computational
timeas shown on table X. On top of that simulation ofroot system
growth can last for a long period oftime, particularly for trees.
Finally, this applicationshows promising results for future
simulations of
h (≈) δt = 10−1 δt = 10−2 δt = 10−31.44 1 s. 7 s. 66 s.
7.18× 10−1 16 s. 46 s. 6 min.3.59× 10−1 70 s. 3.5 min. 27
min.1.79× 10−1 7 min. 17 min. 2 h.8.97× 10−2 50 min. 2h30 9 h.
TABLE XCOMPUTATIONAL TIMES FOR THE SIMULATIONS OF A
ROOT SYSTEM GROWTH PERFORMED (WITH THEPROCESSOR INTEL R©CORETM
I7-7820HQ AT 2.9 GHZ,
RAM 32 GB) BETWEEN t = 1 AND t = 24 MONTHS WITHTHE DG-SPLITTING
ALGORITHM AND THE
CRANK-NICOLSON SCHEME (θ = 1/2).
the root system growth, provided that the compu-tational cost is
not limiting. Further simulationsrequiring much more computational
power has tobe done to check if the convergence is acheived.This
application also point out the difficultiesrelated to the rigorous
simulation validation inrealistic test-cases of root system
growth.
VI. CONCLUSION
In this work, a discontinuous Galerkin approxi-mation method
based on unstructured mesh com-bined with operator splitting has
been described,implemented and tested, to solve an
advection-diffusion-reaction equation used to model thegrowth of
root systems. The code has been val-idated in a simple test case
for which an analyticexpression of the solution is known. The
compu-tations showed that the method convergences witha convergence
rate of two in space with P 1-finiteelements. A convergence rate of
one and two intime were obtained for respectively the implicitEuler
scheme and the Crank-Nicolson schemeboth with and without the
splitting technique. Thecomputations of those convergence rates
requiredthe use of fine mesh grids. For the explicit Eulerscheme,
such fine mesh computations were notperformed since they require
really small timesteps to fulfill the CFL condition, resulting
inhuge additional computational cost. Indeed thecomputational time
of the DG-splitting algorithmbehaves like 1/δt and 1/h2 where δt
and h arerespectively the time step and the mesh size.
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 16 of 19
http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
Similarly, the positivity of the approximatedsolution is
obtained at the expense of the com-putational time since it
requires meshes of smallsize and small time steps. In fact, there
is a CFL-like condition for positivity that has to be fulfilledto
guarantee the positivity of the approximatedsolution. But for a
given mesh size computationsperformed with too small time step can
also leadto a loss of positivity at the beginning of thecomputation
[36]. In that cases, the computationsshowed that there is a
positivity threshold in timeafter which the solution becomes
positive. Thispositivity threshold clearly appeared to diminishwith
the mesh size. This behavior is specific tothe diffusion term. For
the advection term, thecomputations also showed that the positivity
ofthe solution can be preserved, but only at thebeginning of the
simulation and it required a reallysmall mesh size and time step
leading to hugecomputational time. Further studies in terms
ofnumerical analysis has to be done in that direction.
I also performed a more realistic simulation ofroot system
growth. The computations showed thatthe algorithm converged but
additional simulationswith smaller time steps and mesh sizes might
beperformed to recover the full convergence orderand positivity.
Validation of the computation, butabove all the computational time
appeared to bethe major limitations of the root growth
simulationbased on the C-Root model, particularly when itcomes to
deal with trees for which the life span israther a long period of
time. Further improvementson the numerical method has to be done so
thatthe scheme preserves the positivity of the approxi-mated
solution under acceptable CFL conditions interms of computational
time. However, our workshows promising results for the simulation
of theC-Root model which appears to be an appropriatemethodology
for future improvements, like root-soil coupling or nonlinear terms
arising to handlecompetition phenomena.
ACKNOWLEDGMENT
The author would like to thank Y. Dumont(CIRAD, University of
Pretoria) for discussions
and valuable comments about the numericalschemes.
REFERENCES
[1] A. Bonneu, Y. Dumont, H. Rey, C. Jourdan and T. Four-caud, A
minimal continuous model for simulating growthand development of
plant root systems, Plant and Soil,Springer, 2012, 354, 211-22.
https://doi.org/10.1007/s11104-011-1057-7.
[2] E. Tillier and A. Bonneu, Operator splitting for
solvingC-Root, a minimalist and continuous model of rootsystem
growth, Plant Growth Modeling, Simulation, Vi-sualization and
Applications (PMA), 2012 IEEE FourthInternational Symposium on,
2012, 396-402.https://doi.org/10.1109/PMA.2012.6524863.
[3] E. Peynaud, T. Fourcaud and Y. Dumont Numericalresolution of
the C-Root model using DiscontinuousGalerkin methods on
unstructured meshes: applicationto the simulation of roo system
growth, 2016 IEEEInternational Conference on Functional-Structural
PlantGrowth Modeling, Simulation, Visualization and Appli-cations
(FSPMA), IEEE, FSPMA. Qingdao: IEEE, 2016,158-166.
https://doi.org/10.1109/FSPMA.2016.7818302.
[4] H.-G. Roos, M. Stynes and L. Tobiska, Robust
numericalmethods for singularly perturbed differential
equations:convection-diffusion-reaction and flow problems
SpringerScience & Business Media, 2008, 24.
https://doi.org/10.1007/978-3-540-34467-4.
[5] X. Zhang and CW. Shu, Maximum-principle-satisfyingand
positivity-preserving high-order schemes forconservation laws:
survey and new developments.Proceedings of the Royal Society of
London A:Mathematical, Physical and Engineering Sciences, 2011,467,
2752-2776. https://doi.org/10.1098/rspa.2011.0153.
[6] W. Hundsdorfer and J. G. Verwer, Numerical solution
oftime-dependent advection-diffusion-reaction equations.Springer
Science & Business Media, 2013, 33.
DOI10.1007/978-3-662-09017-6
[7] G. Strang, On the construction and comparison ofdifference
schemes. SIAM Journal on Numerical Anal-ysis, SIAM. 5, 506-517.
1968. https://doi.org/10.1137/0705041.
[8] N. Janenko, The method of fractional steps. Springer.1971.
https://doi.org/10.1007/978-3-642-65108-3.
[9] A. Chertock and A. Kurganov, On splitting-basednumerical
methods for convection-diffusion equationsNumerical methods for
balance laws, Aracne Editrice SrlRome, 2010, 24, 303-343.
[10] D. Lanser and J.G. Verwer, Analysis of operatorsplitting
for advection-diffusion-reaction problems fromair pollution
modelling, Journal of computational andapplied mathematics,
Elsevier, 111, 1-2, 1999,
201-216.https://doi.org/10.1016/S0377-0427(99)00143-0.
[11] J. Kačur, B. Malengier, M. Remešı́ková, Convergenceof an
operator splitting method on a bounded domainfor a
convection-diffusion-reaction system, Journal of
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 17 of 19
https://doi.org/10.1007/s11104-011-1057-7https://doi.org/10.1007/s11104-011-1057-7https://doi.org/10.1109/PMA.2012.6524863https://doi.org/10.1109/PMA.2012.6524863https://doi.org/10.1109/FSPMA.2016.7818302https://doi.org/10.1007/978-3-540-34467-4https://doi.org/10.1007/978-3-540-34467-4https://doi.org/10.1098/rspa.2011.0153https://doi.org/10.1137/0705041https://doi.org/10.1137/0705041https://doi.org/10.1007/978-3-642-65108-3https://doi.org/10.1016/S0377-0427(99)00143-0http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
Mathematical Analysis and Applications. 348, 894-914,2008.
https://doi.org/10.1016/j.jmaa.2008.08.017.
[12] M. Remešı́ková, Numerical solution of
two-dimensionalconvection-diffusion-adsorption problems using
anoperator splitting scheme, Applied mathematicsand computation,
184, 116-130, 2007. https://doi.org/10.1016/j.amc.2005.06.018.
[13] J.C. Chrispell, V. Ervin V and E. Jenkins, A fractionalstep
θ-method for convection-diffusion problems, Journalof Mathematical
Analysis and Applications, 333, 204-218,
2007.https://doi.org/10.1016/j.jmaa.2006.11.059.
[14] S. Ganesan and L. Tobiska, Operator-splitting finiteelement
algorithms for computations of high-dimensionalparabolic problems,
Applied Mathematics and Computa-tion, Elsevier, 219, 2013.
6182-6196. https://doi.org/10.1016/j.amc.2012.12.027.
[15] M. Wheeler and C. Dawson, An operator-splittingmethod for
advection-diffusion-reaction problems,MAFELAP Proceedings, 6,
463-482, 1987.
[16] R. Anguelov, C. Dufourd and Y. Dumont, Simulationsand
parameter estimation of a trap-insect model usinga finite element
approach, Mathematics and Computersin Simulation, 2017, 133, 47-75.
https://doi.org/10.1016/j.matcom.2015.06.014.
[17] C. Dufourd and Y. Dumont, Impact of environmentalfactors on
mosquito dispersal in the prospect of sterileinsect technique
control, Computers & Mathematics withApplications, Elsevier,
2013, 66, 1695-1715.
https://doi.org/10.1016/j.camwa.2013.03.024.
[18] V. Girault, B. Rivière and M.F. Wheeler, A splittingmethod
using discontinuous Galerkin for the transientincompressible
Navier-Stokes equations, ESAIM: Math-ematical Modelling and
Numerical Analysis, EDPSciences,39, 1115-1147, 2005.
https://doi.org/10.1051/m2an:2005048.
[19] J. Zhu, Y.T. Zhang, S.A. Newman and M. Alber,Application of
discontinuous Galerkin methods forreaction-diffusion systems in
developmental biology,Journal of Scientific Computing, Springer,
2009, 40, 391-418. https://doi.org/10.1007/s10915-008-9218-4.
[20] N. Ahmed, G. Matthies and L. Tobiska, Finite elementmethods
of an operator splitting applied to populationbalance equations
Journal of Computational and AppliedMathematics, Elsevier. 236,
1604-1621, 2011. https://doi.org/10.1016/j.cam.2011.09.025.
[21] R. Zhang, J. Zhu, A.F Loula and X. Yu, Operatorsplitting
combined with positivity-preservingdiscontinuous Galerkin method
for the chemotaxismodel, Journal of Computational and
AppliedMathematics. 302, 312 - 326, 2016.
https://doi.org/10.1016/j.cam.2016.02.018.
[22] L. Edelstein-Keshet, Mathematical models in biology,SIAM,
46, 1988. ISBN 0-89871-554-7.
[23] B. Perthame, Transport equations in biology.
SpringerScience & Business Media, 2006.
https://doi.org/10.1007/978-3-7643-7842-4.
[24] B. Perthame, Parabolic equations in biology. Growth,
reaction, mouvement and diffusion, Springer, 2015.
https://doi.org/10.1007/978-3-319-19500-1 1
[25] A. Ern, J.L Guermond, Theory and practice of
finiteelements, Springer, 159, 2004.
https://doi.org/10.1007/978-1-4757-4355-5.
[26] V. Volpert, Elliptic Partial Differential Equations:Volume
2: Reaction-Diffusion Equations, Springer, 104,2014.
https://doi.org/10.1007/978-3-0348-0813-2.
[27] D. Kuzmin, A guide to numerical methods fortransport
equations, Friedrich-Alexander-Universitt,Erlangen-Nrnberg,
2010.
[28] F. Brezzi, L. D. Marini and E. Süli, DiscontinuousGalerkin
methods for first-order hyperbolic problems,Mathematical models and
methods in applied sciences,World Scientific, 2004, 14, 1893-1903.
https://doi.org/10.1142/S0218202504003866.
[29] W. H. Reed and T. R. Hill, Triangular mesh methodsfor the
neutron transport equation, Los Alamos ScientificLab., N.
Mex.(USA), report LA-UR-73-479, 1973.
[30] B. Rivière, Discontinuous Galerkin methods forsolving
elliptic and parabolic equations: theory andimplementation, Society
for Industrial and Applied Math-ematics, 2008.
https://doi.org/10.1137/1.9780898717440
[31] J. Oden, I. Babuŝka and C. E. Baumann, Adiscontinuous
hp-finite element method for diffusionproblems, Journal of
computational physics, Elsevier 146,491-519, 1998.
https://doi.org/10.1006/jcph.1998.6032.
[32] D. A. Di Pietro and A. Ern, Mathematical aspectsof
discontinuous Galerkin methods Springer, 69,
2011.https://doi.org/10.1007/978-3-642-22980-0.
[33] X. Zhang and CW. Shu CW.Maximum-principle-satisfying and
positivity-preservinghigh-order schemes for conservation laws:
surveyand new developments, Proceedings of the RoyalSociety of
London A: Mathematical, Physicaland Engineering Sciences, 2011,
467, 2752-2776.https://doi.org/10.1098/rspa.2011.0153.
[34] B. Cockburn and CW. Shu, Runge-Kutta discontinuousGalerkin
methods for convection-dominated problems.Journal of scientific
computing, Springer, 2001, 16, 173-261.
https://doi.org/10.1023/A:1012873910884.
[35] JS. Hesthaven and T. Warburton, Nodaldiscontinuous Galerkin
methods: algorithms,analysis, and applications. Springer, 2007,
54.https://doi.org/10.1007/978-0-387-72067-8.
[36] V. Thomée, On positivity preservation in some
finiteelement methods for the heat equation. International
Con-ference on Numerical Methods and Applications, 2014,13-24.
https://doi.org/10.1007/978-3-319-15585-2 2.
[37] J. Zhu, YT. Zhang, SA. Newman and M. Alber,Application of
discontinuous Galerkin methods forreaction-diffusion systems in
developmental biology.Journal of Scientific Computing, Springer,
2009, 40, 391-418. https://doi.org/10.1007/s10915-008-9218-4.
[38] J.-F. Barczi, H. Rey, S. Griffon and C. Jourdan, DigR:
ageneric model and its open source simulation softwareto mimic
three-dimensional root-system architecture
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 18 of 19
https://doi.org/10.1016/j.jmaa.2008.08.017https://doi.org/10.1016/j.amc.2005.06.018https://doi.org/10.1016/j.amc.2005.06.018https://doi.org/10.1016/j.jmaa.2006.11.059https://doi.org/10.1016/j.amc.2012.12.027https://doi.org/10.1016/j.amc.2012.12.027https://doi.org/10.1016/j.matcom.2015.06.014https://doi.org/10.1016/j.matcom.2015.06.014https://doi.org/10.1016/j.camwa.2013.03.024https://doi.org/10.1016/j.camwa.2013.03.024https://doi.org/10.1051/m2an:2005048https://doi.org/10.1051/m2an:2005048https://doi.org/10.1007/s10915-008-9218-4https://doi.org/10.1016/j.cam.2011.09.025https://doi.org/10.1016/j.cam.2011.09.025https://doi.org/10.1016/j.cam.2016.02.018https://doi.org/10.1016/j.cam.2016.02.018https://doi.org/10.1007/978-3-7643-7842-4https://doi.org/10.1007/978-3-7643-7842-4https://doi.org/10.1007/978-3-319-19500-1_1https://doi.org/10.1007/978-3-319-19500-1_1https://doi.org/10.1007/978-1-4757-4355-5https://doi.org/10.1007/978-1-4757-4355-5https://doi.org/10.1007/978-3-0348-0813-2https://doi.org/10.1142/S0218202504003866https://doi.org/10.1142/S0218202504003866https://doi.org/10.1137/1.9780898717440https://doi.org/10.1006/jcph.1998.6032https://doi.org/10.1007/978-3-642-22980-0https://doi.org/10.1098/rspa.2011.0153https://doi.org/10.1023/A:1012873910884https://doi.org/10.1007/978-0-387-72067-8https://doi.org/10.1007/978-3-319-15585-2_2https://doi.org/10.1007/s10915-008-9218-4http://dx.doi.org/10.11145/j.biomath.2018.12.037
-
Emilie Peynaud, Operator splitting and discontinuous Galerkin
methods for advection-reaction- ...
diversity. Annals of Botany, 2018, 121, 5,
1089-1104,https://doi.org/10.1093/aob/mcy018.
[39] L. X. Dupuy, M. Vignes, An algorithm for thesimulation of
the growth of root systems on deformabledomains. Journal of
Theoretical Biology, 2012, 310, 164-174.
https://doi.org/10.1016/j.jtbi.2012.06.025.
[40] L. Dupuy, P. J. Gregory, A. G. Bengough, Rootgrowth models:
towards a new generation of continuousapproaches. Journal of
experimental botany, Soc Experi-ment Biol, 2010.
https://doi.org/10.1093/jxb/erp389.
[41] P. Bastian, A. Chavarria-Krauser, C. Engwer, W. Jäger,S.
Marnach, M. Ptashnyk, Modelling in vitro growthof dense root
networks Journal of theoretical biology,Elsevier, 2008, 254,
99-109. https://doi.org/10.1016/j.jtbi.2008.04.014.
[42] T. Roose, A. Schnepf, Mathematical models of
plant-soilinteraction Philosophical Transactions of the Royal
Soci-ety of London A: Mathematical, Physical and Engineer-ing
Sciences, The Royal Society, 2008, 366,
4597-4611,https://doi.org/10.1098/rsta.2008.0198.
[43] S. G. Adiku, R. D. Braddock, C. W. Rose,1996, Simulating
root growth dynamics, Environ-mental Software 11 : 99-103.
https://doi.org/10.1016/S0266-9838(96)00041-X.
[44] H. Hayhoe, 1981, Analysis of a diffusion model forplant
root growth and an application to plant soil-wateruptake, Soil
Science 131 : 334-343.
[45] M. Heinen, A. Mollier, P. De Willigen, 2003 Growth ofa root
system described as diffusion numerical model andapplication, Plant
and Soil 252 : 251-265.
https://doi.org/10.1023/A:1024749022761.
[46] V. R. Reddy, Ya. A. Pachepsky, 2001, Testing aconvective
dispersive model of two dimensional root
growth and proliferation in a greenhouse experiment withmare
plants, Annals of Botany 87 : 759-768.
https://doi.org/10.1006/anbo.2001.1409.
[47] P. -H. Tournier, F. Hecht, M. Comte, Finite elementmodel of
soil water and nutrient transport with rootuptake: explicit
geometry and unstructured adaptivemeshing. Transp. Porous Media 106
(2), 487504 (2015).https://doi.org/10.1007/s11242-014-0411-7.
[48] M. Comte, Analysis and Simulation of a Model ofPhosphorus
Uptake by Plant Roots in Current Researchin Nonlinear Analysis: In
Honor of Haim Brezis andLouis Nirenberg, Rassias, T. M. (Ed.),
Springer Inter-national Publishing, 2018, 85-97.
https://doi.org/10.1007/978-3-319-89800-1 4.
[49] F. Gérard, Cé Blitz-Frayret, P. Hinsinger, L.
Pagès,Modelling the interactions between root systemarchitecture,
root functions and reactive transportprocesses in soil Plant and
Soil, 2017, 413,
161-180.https://doi.org/10.1007/s11104-016-3092-x.
[50] H. Brezis, Functional analysis, Sobolev spacesand partial
differential equations. Springer Science& Business Media, 2010.
https://doi.org/10.1007/978-0-387-70914-7.
[51] P. R. Amestoy, I. S. Duff, J. Koster and J.-Y.
L’Excellent,A fully asynchronous multifrontal solver using
distributeddynamic scheduling, SIAM Journal of Matrix Analysisand
Applications, Vol 23, No 1, pp 15-41 (2001).
https://doi.org/10.1137/S0895479899358194.
[52] P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent andS.
Pralet, Hybrid scheduling for the parallel solution oflinear
systems. Parallel Computing Vol 32 (2), pp 136-156 (2006).
https://doi.org/10.1016/j.parco.2005.07.004.
Biomath 7 (2018), 1812037,
http://dx.doi.org/10.11145/j.biomath.2018.12.037 Page 19 of 19
https://doi.org/10.1093/aob/mcy018https://doi.org/10.1016/j.jtbi.2012.06.025https://doi.org/10.1093/jxb/erp389https://doi.org/10.1016/j.jtbi.2008.04.014https://doi.org/10.1016/j.jtbi.2008.04.014https://doi.org/10.1098/rsta.2008.0198https://doi.org/10.1016/S0266-9838(96)00041-Xhttps://doi.org/10.1016/S0266-9838(96)00041-Xhttps://doi.org/10.1023/A:1024749022761https://doi.org/10.1023/A:1024749022761https://doi.org/10.1006/anbo.2001.1409https://doi.org/10.1006/anbo.2001.1409https://doi.org/10.1007/s11242-014-0411-7https://doi.org/10.1007/978-3-319-89800-1_4https://doi.org/10.1007/978-3-319-89800-1_4https://doi.org/10.1007/s11104-016-3092-xhttps://doi.org/10.1007/978-0-387-70914-7https://doi.org/10.1007/978-0-387-70914-7https://doi.org/10.1137/S0895479899358194https://doi.org/10.1137/S0895479899358194https://doi.org/10.1016/j.parco.2005.07.004http://dx.doi.org/10.11145/j.biomath.2018.12.037
IntroductionThe modelModelling root growth with PDE: the C-Root
modelThe weak problemThe positivity preserving property of the
solution
Approximation of the modelThe operator splitting techniqueThe
advection step: DG upwind schemeThe diffusion stepThe reaction
step
Validation of the splitting algorithm with a simple test
caseDescription of the simple test-caseNumerical validation and
convergenceSome comments on the positivityPositivity of the full
problemPositivity of the pure diffusion problemPositivity of the
pure advection problem
Application to the simulation of root system growthThe C-Root
parameters for Eucalyptus root growthSome simulations
ConclusionReferences