Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Converging entropy-diminishing finite volume scheme forthe Stefan-Maxwell model
Clément Cancès2,3, Virginie Ehrlacher1,2, Laurent Monasse2,4
1Ecole des Ponts ParisTech
2INRIA
3Université de Lille
4Université Côte d’Azur
LJLL seminar, 19th of June 2020 1 / 40
Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Outline of the talk
Introduction to cross-diffusion systems
The Stefan-Maxwell system
Finite volume scheme
Numerical results
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Outline of the talk
Introduction to cross-diffusion systems
The Stefan-Maxwell system
Finite volume scheme
Numerical results
3 / 40
Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Multi-species systems
• Population dynamics (zoology, epidemiology...)• Materials science (atomic diffusion, gas mixtures...)• Tumor growth
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Modeling of multi-species systems
In this talk, we will more specifially focus on multi-species systems arising inmaterials science, typically for the modelisation of atomic diffusion withinsolids or gas mixtures.
Different types of models may be used to model such systems, at differentscales:• Particle models: for instance using Newton’s laws with interactions
between species• Markov chains: space is discretized with a grid and species move to
neighboring cells.• Stochastic differential equations: dynamics defined on a continuous
state space with stochastic noise, like Brownian motion• Kinetic equations: distribution function which depends on space,
velocity...• Here: Diffusive equations for concentrations or volumic fractions.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Diffusion systems
Let us consider a system composed of n different species occupying abounded domain Ω ⊂ Rd . For all 1 ≤ i ≤ n, let us denote by ui (t , x) thevolumic fraction of the i th species at point x ∈ Ω and time t > 0.
General form of a diffusion system:
∂tui − div (Ji ) = 0, ui (t = 0, ·) = u0i ,
with no-flux boundary conditions on ∂Ω, and Ji (t , x) ∈ Rd the flux of the i thspecies at point x and time t > 0.
Fick’s law: Ji = Di∇ui for some Di > 0. This leads to a system of decoupleddiffusion equations.
Fick’s law is not always valid and in general Ji may depend on ∇u1, · · · ,∇unin multicomponent systems.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Cross-diffusion systems
In general,
∀1 ≤ i ≤ n, Ji (t , x) =n∑
j=1
Aij (u)∇uj , (1)
where Aij : Rn → R is a smooth function for all 1 ≤ i , j ≤ n.
Equations (1) can be rewritten in a more condensed form using the notation
u = (u1, · · · , un), J = (J1, · · · , Jn)
asJ = A(u)∇u
where for all u ∈ Rn, A(u) = (Aij (u))1≤i,j≤n ∈ Rn×n is called the diffusion
matrix of the system.
General form of a cross-diffusion system:
∂tu − div (A(u)∇u) = 0, u(t = 0, ·) = u0 = (u01 , · · · , u0n)
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Cross-diffusion systems as hydrodynamic limits
Hydrodynamic limits of microscopic and mesoscopic models lead tocross-diffusion systems with non-diagonal diffusion matrices:• Markov chains on discrete state space: Quastel 1991; Erignoux 2018; ...• Continuous stochastic differential equations: Chen, Daus, Jüngel 2019;
...• Kinetic equations: Boudin, Grec, Salvarini, 2015; Boudin, Grec, Pavant,
2017; Bondesant, Briant, 2019; ...
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Outline of the talk
Introduction to cross-diffusion systems
The Stefan-Maxwell system
Finite volume scheme
Numerical results
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Example: the Stefan-Maxwell system• Proposed by Maxwell 1866/Stefan 1871.• Models the evolution of a gas mixture in non dilute regime with n
components. The functions u1, · · · , un model the volumic fractions of thegas components: from a modelling point of view, it holds that for all t > 0and x ∈ Ω,∀1 ≤ i ≤ n, ui (t , x) ≥ 0 and
∑ni=1 ui (t , x) = 1.
• Duncan-Toor 1962: Comparison between the Stefan-Maxwell model andexperimental measurements for a system composed of hydrogen,nitrogen and carbon dioxide.
• Boudin, Grec, Salvarini, 2015: derivation from the Boltzmann equationfor simple mixtures.
• Application: Patients with airways obstruction inhale Heliox to speed updiffusion
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
The Stefan-Maxwell systemThe Stefan-Maxwell system reads, together with appropriate initial andno-flux boundary conditions,
∂tui − div (Ji ) = 0,∇ui +
∑nj=1 Bij (u)Jj = 0,∑n
i=1 Ji = 0
where
∀1 ≤ i 6= j ≤ n, Bij (u) = −cijui , Bii (u) =∑
1≤j 6=i≤n
cijuj
withcij = cji > 0.
Notation:〈u, v〉 :=∑n
i=1 uivi for all u := (ui )1≤i≤n, v := (vi )1≤i≤n ∈ Rn.
Condensed form: ∂tu − div (J) = 0,∇u + B(u)J = 0,〈1, J〉 = 0
(2)
where 1 = (1, · · · , 1).11 / 40
Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Properties of the matrix B(u)
Giovangigli, 1999; Bothe, 2011; Boudin, Grec, Salvarani, 2012; Jüngel,Steltzer, 2013...
A :=
{u := (ui )1≤i≤n ∈ Rn+,
n∑i=1
ui = 〈1, u〉 = 1
}
V :=
{v := (vi )1≤i≤n ∈ Rn,
n∑i=1
vi = 〈1, v〉 = 0
}
Lemma (Jüngel, Steltzer, 2013)Let u ∈ (R∗+)n ∩ A. Then, it holds that
Span B(u) = V and Ker B(u) = Span{u}.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Properties of the matrix B(u)Consequence: Thus, for any u ∈ (R∗+)n ∩ A, for any vector z ∈ V and anyvector y ∈ Rn such that 〈y , u〉 6= 0, there exists a unique solution x ∈ Rnsolution to
B(u)x = z and 〈y , x〉 = 0.
Assume now that there exists a solution (u, J) to (2) such thatu(t , x) ∈ (R∗+)n ∩ A for almost all t > 0 and x ∈ Ω. Then, ∇u(t , x) ∈ Vd since
n∑i=1
ui = 1 a.e. implies thatn∑
i=1
∇ui = 0 a.e.
Besides, 〈1, u〉 = 1 6= 0 a.e. Then, a.e., there exists a unique solutionJ(t , x) ∈ Rn×d such that, a.e.{
B(u)J +∇u = 0,〈1, J〉 = 0,
and there exists a matrix field A : (R∗+)n ∩ A → Rn×n such that
J = A(u)∇u.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Weak solution for the Stefan-Maxwell system
Let T > 0 be some final time and QT := (0,T )× Ω.
Definition (Weak solution)A weak solution (u, J) to the the Stefan-Maxwell system, corresponding tothe initial profile u0 ∈ L∞(Ω;A), with no-flux boundary conditions, is a pair(u, J) such that u ∈ L∞(QT ;A) ∩ L2((0,T ); H1(Ω)n), ∇
√u ∈ L2(QT )n×d ,
J ∈ L2(QT ;Vd ) satisfies
B(u)J +∇u = 0 a.e. in QT
and such that for all φ := (φi )1≤i≤n ∈ C∞c ([0,T )× Ω)n,
−∫ ∫
QT
〈u, ∂tφ〉+∫
Ω
〈u0, φ(0, ·)〉+∫ ∫
QT
n∑i=1
Ji · ∇φi = 0.
Theorem (Jüngel, Steltzer, 2013)There exists at least one weak solution to (2) in the sense of the previousdefinition.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Entropic structure of cross-diffusion systemsA key ingredient in the proof of the existence of a global in time solution tomany cross-diffusion systems is the fact that some of these systems enjoy anentropic structure.
∂tu − div (A(u)∇u) = 0
More precisely, for many cross-diffusion systems (including theStefan-Maxwell system), there exists an entropy functional which is aLyapunov function for the system, and enables to obtain appropriateestimates in order to establish the existence of a global in time weak solution.
In general, such an entropy functional reads as E(u) =∫
Ωh(u) for some
convex function h : A → R such that D2h(u)A(u) is a positive definite matrixfor all u ∈ A ∩ (R∗+)n.
ddt
∫Ω
h(u) =∫
Ω
Dh(u) · ∂tu = −∫
Ω
∇Dh(u) · A(u)∇u
= −∫
Ω
∇u · D2h(u)A(u)∇u ≤ 0.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Entropy dissipation for the Stefan-Maxwell systemFor the Stefan-Maxwell system,
h(u) =n∑
i=1
ui log ui
Let c∗ := min1≤i 6=j≤n cij > 0, c ij = cij − c∗ ≥ 0 and c := max1≤i 6=j≤n c ij .
Then, the following inequality holds for all u solution to the Stefan-Maxellsystem
ddt
E(u) ≤ −12α
n∑i=1
∫Ω
|∇√
ui |2 −12
c∗∫
Ω
|J|2 ≤ 0, (3)
withα :=
4c∗ + 2c
> 0.
This inequality enables to obtain bounds on∫ ∫QT
|∇√
ui |2 and∫ ∫
QT
|J|2
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Outline of the talk
Introduction to cross-diffusion systems
The Stefan-Maxwell system
Finite volume scheme
Numerical results
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Numerical scheme for the Stefan-Maxwell system: wishlist
• the non-negativity of the volumic fractions;• the conservation of mass∫
Ω
ui (t , ·) =∫
Ω
u0i , ∀1 ≤ i ≤ n.
• the preservation of the volume-filling constraint
n∑i=1
ui = 1 a.e.
• the entropy dissipation relation (3) (or a discrete version of it).
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Numerical schemes for the Stefan-Maxwell system: literature
Numerical methods for cross-diffusion systems preserving thesemathematical properties is a very active field of research!
Burger, Cancès, Carillo, Chainais-Hillaret, Daus, Filbet, Guichard, Jüngel,Pietschamnn, Schmidtchen...
In the particular case of the Stefan-Maxell system,• Boudin, Grec, Salvarani, 2012: ternary system, dimension 1• Jüngel, Leingang, 2019: finite element approximation
Here, a finite volume scheme based on a two-point flux approximation,inspired from [Cancès, Gaudeul, 2020] where the authors considered a moresimple cross-diffusion system.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Fundamental remark for the scheme
For all u ∈ Rn+, it holds that
B(u) = c∗〈1, u〉I + c∗C(u) + B(u)
where, for all 1 ≤ i , j ≤ n,
Cij (u) = ui , B ii (u) =∑
1≤i 6=j≤n
c ijuj , B ij (u) = −c ijui i 6= j
The matrix B has the same expression as B except that the coefficients cijare replaced by c ij .
In particular, if u ∈ A, B(u) = c∗I + c∗C(u) + B(u). Moreover, for all J ∈ V,C(u)J = 0. Thus,
B(u)J = c∗J + B(u)J
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Admissible mesh
T : set of cells E : set of faces (or edges) (xK )K∈T : set of cell centers
An admissible mesh of Ω is a triplet (T , E , (xK )K∈T ) such that the followingconditions are fulfilled:
(i) Each cell K ∈ T is non-empty, open, polyhedral and convex.
K ∩ L = ∅, K 6= L,⋃
K∈T
K = Ω
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Admissible meshT : set of cells E : set of faces (or edges) (xK )K∈T : set of cell centers
An admissible mesh of Ω is a triplet (T , E , (xK )K∈T ) such that the followingconditions are fulfilled:(ii) Each face σ ∈ E is closed and contained in an hyperplane of Rd and
such that its d − 1-dimensional measure mσ := Hd−1(σ) is positive.For all K ∈ T , there exists a subset EK ⊂ E such that
⋃σ∈EK
σ = ∂K .Besides, E =
⋃K∈T EK .
For all K 6= L ∈ T , either K ∩ L is equal to a single face σ ∈ E (and inthis case we denote by σ = K |L), or the d − 1-dimensional Hausdorffmeasure of K ∩ L is 0.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Admissible mesh
T : set of cells E : set of faces (or edges) (xK )K∈T : set of cell centers
An admissible mesh of Ω is a triplet (T , E , (xK )K∈T ) such that the followingconditions are fulfilled:
(iii) Orthogonality condition: The cell centers (xK )K∈T satisfy xK ∈ K , andare such that, if K , L ∈ T share a face σ = K |L ∈ E , then the vectorxK − xL is orthogonal to the face K |L.
Balaven, Bennis, Boissonat, Yvinec, 2006: construction of orthogonalmeshes.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Time discretization
In the rest of the talk, quantities defined on this discrete mesh will be denotedby bold symbols.
Let ∆t > 0, tp = p∆t for all p ∈ N and PT ∈ N∗ such that tPT = PT ∆ = T .
The numerical method is an iterative scheme, where for all p ∈ N∗, a discretesolution
up := (upi )1≤i≤n ∈(RT)n,
so that upi =(
upi,K)
K∈T∈ RT with
upi,K an approximation of the function ui at time tp in the cell K ,
will be computed given the value of the discrete solution at the previous timestep up−1.
Let u0 = (u0i )1≤i≤n ∈(RT)n be a discretized initial condition.
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Notation• For all K ∈ T , mK = |K | the Lebesgue measure of the cell K ;• For all σ ∈ E , mσ = Hd−1(σ) the d − 1-dimensional Hausdorff measure
of the face σ,
dσ :={|xK − xL| if σ = K |L is an interior face;d(xK , σ) if σ ∈ EK is an exterior face,
andτσ =
mσdσ
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NotationFor all v = (vK )K∈T ∈ RT , for all K ∈ T and all σ ∈ EK , we denote by vKσ themirror value of vK across σ, i.e.
vKσ ={
vL if σ = K |L for some L ∈ T ,vK if σ is an exterior face,
The oriented jump of v across σ is defined by
DKσv := vKσ − vK
Finally, vσ,log denotes the logarithmic mean between vK and vKσ, i.e.
vσ,log :=
0 if min(vK , vKσ) ≤ 0,vK if vK = vKσ ≥ 0,
vK−vKσlog(vK )−log(vKσ)
otherwise.
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Numerical scheme
For all K ∈ T and all 1 ≤ i ≤ n,
mKupi,K − u
p−1i,K
∆t+∑σ∈EK
mσJpi,Kσ = 0, (4)
where for all σ ∈ EK , JpKσ =(
Jpi,Kσ)
1≤i≤n∈ Rn is computed as follows:
• if σ = K |L is an interior face,
1dσ
DKσupi + c∗Jpi,Kσ +
∑1≤j≤n
B ij (upσ,log)J
pj,Kσ = 0, ∀1 ≤ i ≤ n, (5)
where upσ,log =(
upi,σ,log)
1≤i≤n, and
JLσ = −JKσ (6)
• if σ is an exterior face,JpKσ = 0. (7)
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Properties of the scheme
Theorem (Cancès, VE, Monasse, 2020)Let (T , E, (xK )K∈T ) be an admissible mesh of Ω and let u0 be an initial condition suchthat u0 ∈ AT . Then, for all p ∈ N∗, the nonlinear system of equations (4)-(5)-(6)-(7)has at least a (strictly) positive solution up ∈ AT . This solution satisfies∑
K∈Tupi,K =
∑K∈T
u0i,K .
In addition, the corresponding fluxes Jp =(JpKσ
)σ∈E are uniquely determined by
(5)-(6)-(7) and belong to VE , i.e.
∀K ∈ T , ∀σ ∈ EK ,n∑
i=1
Jpi,Kσ = 0.
Moreover, the following discrete entropy dissipation estimate holds
ET (up) + ∆t∑
σ=K |L∈Eint
(c∗
2mσdσ |JpKσ |
2 +α
2τσ |DKσ
√up|2
)≤ ET (up−1)
where the discrete entropy functional is defined as
ET (u) =∑
K∈T
n∑i=1
mK ui,K log(ui,K ), ∀u = (ui )1≤i≤n ∈ AT .
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Reconstruction of approximate densities
Consider the particular initial condition u0 ∈ AT defined by
u0i,K =1
mK
∫K
u0i ,
and let (up)p∈N∗ be a sequence of solutions of the scheme satisfying theconditions of the theorem. Let also (Jp)p∈N∗ be the sequence ofcorresponding fluxes.
For all 1 ≤ i ≤ n, let ui,T ,∆t : QT → R be the piecewise constant functiondefined by
ui,T ,∆t (t , x) = upi,K , ∀x ∈ K , ∀t ∈ (tp−1, tp].
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Reconstruction of approximate gradients and fluxes: diamond cell
The half diamond cell ∆Kσ associated to K ∈ T and σ ∈ EK is defined as theconvex hull of xK and σ.
For all σ ∈ E , we define the diamond cell associated to σ as
∆σ :=
{∆Kσ ∪∆Lσ if σ = K |L is an interior face,∆Kσ if σ ∈ EK is an exterior face.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Reconstruction of approximate gradients and fluxesFor all K ∈ T and σ ∈ EK , denoting by xσ the orthogonal projection of xK ontoσ, we denote by
nKσ =
{xL−xK
dσif σ = K |L is an interior face,
xσ−xKdσ
if σ ∈ EK is an exterior face.
For all 1 ≤ i ≤ n, let Ji,E,∆t : QT → Rd be the piecewise constant functiondefined by
Ji,E,∆t = dJpi,KσnKσ, ∀x ∈ ∆σ, σ ∈ EK , t ∈ (tp−1, tp].
Let us also define ∇E,∆tu i : QT → Rd and ∇E,∆t√
u i : QT → Rd thepiecewise constant functions defined by
∇E,∆tu i = dDKσupi nKσ, ∀x ∈ ∆σ, σ ∈ EK , t ∈ (tp−1, tp],
and
∇E,∆t√
u i = dDKσ√
upi nKσ, ∀x ∈ ∆σ, σ ∈ EK , t ∈ (tp−1, tp].
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What about convergence?
Let (Tm, Em, (xmK )K∈Tm )m∈N be a sequence of admissible meshes such that
hTm := maxK∈Tm
diam(K ) −→m→+∞
0
andζTm := min
K∈Tmminσ∈EK
d(xK , σ)dσ
≥ η, ∀m ∈ N,
for some η > 0 independent of m.
Let (∆tm)m∈N be a sequence of positive time steps such that ∆tm −→m→+∞
0.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Convergence of the scheme
Theorem (Cancès, VE, Monasse, 2020)There exist u ∈ L∞(QT ;A) ∩ L2((0,T ); H1(Ω)n) with
√u ∈ L2((0,T ); H1(Ω)n)
and J ∈ L2(QT ,Vd ) such that, up to the extraction of a subsequence,
uTm,∆tm = (ui,Tm,∆tm )1≤i≤n −→m→+∞ u a.e. in QT ,
∇Em,∆tm√
u =(∇Em,∆tm
√u i)
1≤i≤n −→m→+∞∇√
u weakly in L2(QT )n×d ,
∇Em,∆tm u = (∇Em,∆tm u i )1≤i≤n −→m→+∞∇u weakly in L2(QT )n×d ,
JEm,∆tm = (Ji,Em,∆tm )1≤i≤n −→m→+∞ J weakly in L2(QT )n×d .
Besides, (u, J) is a weak solution of the Stefan-Maxwell problem.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Outline of the talk
Introduction to cross-diffusion systems
The Stefan-Maxwell system
Finite volume scheme
Numerical results
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
One-dimensional test case: initial concentration profiles
3 species, uniform discretization with 150 cells, ∆t = 10−5.
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
One-dimensional test case3 species, uniform discretization with 150 cells, ∆t = 10−5.
test test
test
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
One-dimensional test case: decay of the entropy
3 species, uniform discretization with 150 cells, ∆t = 10−5.
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Two-dimensional test case3 species, uniform discretization with 50× 50 cells, ∆t = 5.10−5.
test test
test
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Conclusions
• Finite volume scheme for the Stefan-Maxwell model using Two-PointFlux Approximation
• Arbitrary dimension and number of species• Preserves the non-negativity of the volumic fractions, the volume-filling
constraint, the conservation of mass, and a discrete version of theentropy dissipation inequality.
• Provably converging.
Open questions and perspectives:• Rates of convergence?• Finite volume schemes for general cross-diffusion systems with entropic
structure?• Numerical methods for cross-diffusion systems on moving domains?
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Introduction to cross-diffusion systems The Stefan-Maxwell system Finite volume scheme Numerical results
Merci pour votre attention!
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