converging entropy-diminishing finite volume scheme for ... · introduction to cross-diffusion...

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


Outline of the talk

Introduction to cross-diffusion systems

The Stefan-Maxwell system

Finite volume scheme

Numerical results

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Outline of the talk




Numerical results

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Multi-species systems

• Population dynamics (zoology, epidemiology...)• Materials science (atomic diffusion, gas mixtures...)• Tumor growth

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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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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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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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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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Outline of the talk




Numerical results

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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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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


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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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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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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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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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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Outline of the talk




Numerical results

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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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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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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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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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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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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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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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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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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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Outline of the talk




Numerical results

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One-dimensional test case: initial concentration profiles

3 species, uniform discretization with 150 cells, ∆t = 10−5.

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One-dimensional test case3 species, uniform discretization with 150 cells, ∆t = 10−5.

test test

test

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anim_u1_1D.mp4Media File (video/mp4)

Lavf58.20.100


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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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Lavf58.20.100


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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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Merci pour votre attention!

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Introduction to cross-diffusion systemsThe Stefan-Maxwell systemFinite volume schemeNumerical results

converging entropy-diminishing finite volume scheme for ... · introduction to cross-diffusion...

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