Simulation of a time dependent advection-reaction-diffusion ......0 is the given initial solution....

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


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




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


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.




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)




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




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 ìph (t; v) is given by ìph (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

=

ìph (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




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-




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.




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




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


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


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


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


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




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


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


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


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


TABLE VIIMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED

FOR THE PURE DIFFUSION PROBLEM WITH THECRANK-NICOLSON SCHEME (TOP) AND THE IMPLICIT


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




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


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


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


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


TABLE IXMINIMUM VALUE OF THE DOF (mini,kXki ) COMPUTED

FOR THE PURE ADVECTION PROBLEM WITH THECRANK-NICOLSON SCHEME (TOP) AND THE IMPLICIT


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




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




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.




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


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


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


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


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.


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

Date post:	31-Jan-2021
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Simulation of a time dependent advection-reaction-diffusion ......0 is the given initial solution....

Documents