Un schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme

Transcription

1 Un schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme Christophe Berthon, Anaïs Crestetto et Françoise Foucher LMJL, Université de Nantes ANR GEONUM Séminaire de l équipe EDP Analyse Numérique Nice, 8 juin 25 Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme /39

2 Main motivations Problem and objectives Hyperbolic systems of conservation laws with source term t w + x F (w) = S (w), w Ω. ν Construct Finite Volume schemes with some good properties. Steady states: w Ω such that x F (w) = ν S (w) exact capture of some of them. Preservation of asymptotic diffusive regime: ν. Other properties of the solution (positivity of densities, etc.). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 2/39

3 of chemotaxis,2 where t ρ+ x (ρu) =, () t (ρu)+ x ( ρu 2 + p ) = χρ x φ αρu, (2) - ρ(x,t) : particles density, - u(x,t) R: mean velocity, t φ D xx φ = aρ bφ, (3) - φ(x, t) : concentration of chemoattractant, - p(ρ) = ερ γ : pressure law, with γ > adiabatic exponent and ε > a constant, - χ, α, D >, a > and b > some parameters. Natalini, Ribot, Twarogowska, CMS Twarogowska, PhD Thesis 2. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 3/39

4 Objectives Problem and objectives Preserve equilibrium states (at rest, with u = ), that are given by { ε γ χ γ ργ φ = K, D xx φ = aρ bφ, with a constant K. Exact capture when γ = 2. W-B scheme on the hyperbolic part ()-(2),2, using a suitable extension of the hydrostatic reconstruction 3, and also on the equation (3) for φ. Deal with vacuum regions. Natalini, Ribot, Twarogowska, CMS Twarogowska, PhD Thesis 2. 3 Audusse, Bouchut, Bristeau, Klein, Perthame, SIAM JSC 24. (4) Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 4/39

5 Chosen strategy Problem and objectives Derive a finite volume scheme of Godunov-type. Construct a suitable approximate Riemann solver w: - that verifies the consistency condition introduced by Harten, Lax and van Leer 4, - by imposing the preservation of steady states of interest (with a good approximation of the source term 5 and using their explicit expression), - by using a cut-off procedure 6 when dealing with vanishing densities. 4 Harten, Lax, van Leer, SIAM review Berthon, Chalons, to appear in Math. of Comp. 6 Audusse, Chalons, Ung, CMS 25. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 5/39

6 Outline Problem and objectives Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 6/39

7 Problem and objectives Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity We first look at the two first equations ()-(2), that we rewrite as with ( ρ w = ρu ) (, F (w) = t w + x F (w) = S (w) (5) ρu ρu 2 + p considering φ as a known source term. ) ( and S (w) = χρ x φ αρu ), First study: without friction (α = ). Second study: with friction (α > ). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 7/39

8 Approximate Riemann solver Approximate Riemann solver w( x t,w L,w R ) defined by: velocities λ L < < λ R : Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity λ L = min(,λ L,λ R ) and λ R = max( +,λ + L,λ+ R ), where λ ± L and λ± R denote the eigenvalues of the flux Jacobian matrix: λ ± = u ± c where c = c(ρ) = P (ρ) = εγρ γ, intermediate states wl and w R (will be defined later), t the CFL condition: x max( λ L, λ R ) 2. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 8/39

9 HLL consistency condition Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity Consistency condition from Harten, Lax and van Leer: x 2 w( x x x t,w L,w R )dx = w R ( x x 2 }{{} x t,w L,w R )dx, 2 }{{} Ã x 2 with w R ( x t,w L,w R ) the exact solution of the Riemann problem. Left side: Ã = 2 (w L + w R )+ t x (λ L(w L w L )+λ R (w R w R)). A R Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 9/39

10 t w + x F (w) = S (w) (5) Right side (by integrating on [, t] [ x equation (5)) in the case α = : Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity 2, x 2 ] the exact A R = 2 (w L + w R ) t x (F (w R) F (w L ))+ t Exact source term S R = t x 2 t x χρ x R x φdxdt 2 approximated by S. Four other unknowns: w L = ( ρ L We need 5 equations. ρ L u L ) ( ). S R and w R = ( ρ R ρ R u R ). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme /39

11 Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity Two first equation: Ã = A R λ L (ρ L ρ L )+λ R (ρ R ρ R) = ρ L u L ρ R u R, λ L (ρ L u L ρ L u L )+λ R (ρ R u R ρ Ru R ) = ρ L u 2 L + p L ρ R u 2 R p R + xs. Third equation: ρ L u L = ρ R u R =: q (continuity of momentum). Two last equations: study the steady states at rest. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme /39

12 Steady states at rest Problem and objectives Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity Steady states at rest associated to (5) given by: { u =, e χφ = K, where e(ρ) is defined by x e = ρ xp which is, since P(ρ) = ερ γ : γ e(ρ) = ε γ ργ + e with e an arbitrary constant. In the Riemann problem: { ul = u R =, e L χφ L = e R χφ R = K. Approximated Riemann solver preserves steady states at rest: { u L = u R = ( q = ), e L χφ L = e R χφ R = K. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 2/39

13 Source-term approximation 5 Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity We are at steady state à = A R becomes { λ L (ρ L ρ L )+λ R(ρ R ρ R) =, (λ R λ L )q + p R p L = x S. We want to preserve steady states we have to ensure q = we suggest to put the following consistent expression of S : Fourth equation. S = χ x p R p L e R e L (φ R φ L ). Necessary condition for Well-Balanced property. 5 Berthon, Chalons, to appear in Math. of Comp. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 3/39

14 Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity For the last equation, we choose to impose e L ρ L ρ L χφ L = e R ρ R ρ R χφ R, which is consistent with e R e L = φ R φ L and gives a linearization of the equation x e = χ x φ. Sufficient condition for Well-Balanced property. Solving the system of five equations expressions for ρ L, ρ R, q and S. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 4/39

15 Case with friction: α > Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity What changes? The second equation: t (ρu)+ x ( ρu 2 + p ) = χρ x φ αρu. Integrating on [, t] [ x 2, x ] 2 gives F ( t) = 2 (ρ Lu L +ρ R u R ) t ( ρr ur 2 x +p R ρ L ul 2 p ) t L + tsr α F (t)dt, where Equation in F ( t): F (t) = x/2 (ρu) x R (x,t)dx. x/2 F ( t) = ( ρr ur 2 x + p R ρ L ul 2 p ) L + SR αf ( t). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 5/39

16 Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity Solution: F ( t) = Ke α t + α with K = 2 (ρ Lu L +ρ R u R ) α [ ] ( ρr ur 2 x + p R ρ L ul 2 p ) L + SR, [ ] ( ρr ur 2 x + p R ρ L ul 2 p ) L + SR. HLL consistency condition: Ã2 = F ( t). Approximation of S R in order to preserve equilibrium (u = ): S defined previously suits. ρ L and ρ R not changed. q is the only quantity that has to be modified when taking α >. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 6/39

17 Correction for the positivity Construction of the approx. Riemann solver without friction Construction of the approx. Riemann solver with friction Correction for the positivity Min-Max procedure 6 in order to ensure ρ L and ρ R, where ρ L = ρ λ R R ρ L u L ρ R u R L + δ R, δ R λ L δ L λ R δ R λ L δ L λ R ρ R = ρ R + λ L R δ R λ L δ L λ R δ L ρ L u L ρ R u R δ R λ L δ L λ R, with δ R := e R ρr, δ L := e L ρ L and R := χ(φ R φ L ) e R. For simplicity reasons: λ R = λ L. Consistency gives now: Ã = A R, ρ L +ρ R = ρ L +ρ R ρ Ru R ρ L u L λ R =: 2ρ HLL. We take: ρ R = min(max(,ρ R ),2ρ HLL), ρ L = min(max(,ρ L ),2ρ HLL). 6 Audusse, Chalons, Ung, CMS 25. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 7/39

18 Problem and objectives Construction of the approx. Riemann solver Full numerical scheme W-B property AP property We now look at the equation (3) on φ, that we rewrite as: t φ D x ψ = aρ bφ, where ρ is assumed to be known (since computed previously) and ψ := x φ. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 8/39

19 HLL consistency condition Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Consistency condition from Harten, Lax and van Leer: x/2 ( x ) φ x x/2 t,φ L,φ R dx }{{} Ã = x x/2 x/2 ( x φ R t,φ L,φ R )dx, } {{ } A R with φ R the exact Riemann solver and φ the approximated one. Left side: Ã = 2 (φ L +φ R )+ t x (λ L(φ L φ L )+λ R (φ R φ R)). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 9/39

20 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property t φ D x ψ = aρ bφ (3) Right side (by integrating on [, t] [ x equation (3)): A R = 2 (φ L +φ R )+ D t x (ψ R ψ L )+ t x First part of the model: x We get: x/2 x/2 2, x 2 x/2 x/2 ] the exact aρ R bφ R dxdt. ρ R (x,t) dx = 2 (ρ L +ρ R ) t x (ρ Ru R ρ L u L ). G( t) := A R = 2 (φ L +φ R )+ D t x (ψ R ψ L ) ( ) t +a 2 (ρ L +ρ R ) t2 2 x (ρ Ru R ρ L u L ) t b G(t)dt. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 2/39

21 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Equation on G( t): G ( t)+bg( t) = D x (ψ R ψ L )+a Solution: ( 2 (ρ L +ρ R ) t ) x (ρ Ru R ρ L u L ). G( t) = 2 (φ L +φ R ) e b t +β t +β 2 ( e b t) with β = a b x (ρ Ru R ρ L u L ) and β 2 = D b x (ψ R ψ L )+ a 2b (ρ L +ρ R )+ a b 2 x (ρ Ru R ρ L u L ). Consistency condition: Ã = A R = G( t). + Choice: φ L φ L = φ R φ R. Expressions for φ L and φ R. And what about ψ L and ψ R? Answer later... Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 2/39

22 Finite Volumes scheme Construction of the approx. Riemann solver Full numerical scheme W-B property AP property D domain [,L] discretized by N + points: x i = i x, i =,...,N, x = L N. Evolution in time of w n i x w n+ i φ n i x t n = n t for a time step t. xi+/2 x w (x,t n ) dx and i /2 xi+/2 x φ(x,t n )dx, where i /2 Scheme coming from the previously defined Riemann solvers: ( ( ) ( )) = wi n t x λ i 2,R wi n w i 2,R λ i+ 2,L wi n w i+ 2,L, ( )) = φ n i t x φ n i φ ) λ i 2,R i+ φ n i φ i+ 2,L. φ n+ i λ i 2,R ( 2,L ( Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 22/39

23 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Definition of ψ n i ψi n such that good approximation of ( x φ) n i, for example of the form ψi n = ( φ n 2 x i+ φ n ) i ( x) where ( x) has to be consistent with when x tends to. ( x): correction term such that φ n+ i = φ n i at equilibrium in the particular case γ = 2. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 23/39

24 Equilibrium for γ = 2 Problem and objectives Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Equilibrium given by: u =, φ = 2ε χ ρ+k, D xx φ bφ = aρ, { u =, xx ρ χ 2εD ( 2εb χ a ) ρ = Kb χ 2εD. Solutions,2 : ( if ρ = : φ(x) = Acosh x ) ( b/d + B sinh x ) b/d, ( if ρ >, C < : φ(x) = Acos x ) ( C + B sin x ) C φ p, ρ(x) = χ (φ(x) K), 2ε ( if ρ >, C > : φ(x) = Acosh x ) ( C + B sinh x ) C φ p, ρ(x) = χ (φ(x) K), 2ε where A and B are some constants, C = ( D b aχ) 2ε and φp = Kaχ 2εb aχ. Natalini, Ribot, Twarogowska, CMS Twarogowska, PhD Thesis 2. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 24/39

25 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Injecting these solutions in the Finite Volumes scheme and imposing φ n+ i = φ n i gives the appropriate expression of ( x): if ρ = : ( x) = x2 2 if ρ >, C < : ( x) = x2 2 if ρ >, C > : ( x) = x2 2 Expression of ψ n i b/d ( b/d x ), cosh ( C cos C x ), C cosh( C x). ψi n = ( φ n 2 x i+ φ n ) i ( x) with this ( x) steady states exactly preserved for γ = 2 and approximated for γ > 2. Min-Max procedure in order to ensure φ. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 25/39

26 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Asymptotic regime: t, α, u Rescaling u = νu ν, t = tν ν, α = αν ν, with ν > parameter devoted to tend to zero. Rescalled system given by ν t νρ ν +ν x (ρ ν u ν ) =, ν 2 t ν (ρ ν u ν ( )+ x ν 2 ρ ν (u ν ) 2 + p(ρ ν ) ) = χρ ν x φ ν α ν ρ ν u ν, ν t νφ ν D xx φ ν = aρ ν bφ ν. Chapman-Enskog expansion: ρ ν = ρ +O(ν), ρ ν u ν = ρ u +O(ν), φ ν = φ +O(ν). Zero-order governed by t νρ ( + x χ/α ν ρ x φ /α ν x p(ρ ) ) =, D xx φ = bφ aρ, ρ u = /α ν x p(ρ )+χ/α ν ρ x φ. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 26/39

27 Numerical framework Problem and objectives Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Scheme: w n+ i φ n+ i = w n i t x = φ n i t x Rescalling: ( ( Simplification: ( λ i 2,R ( λ i 2,R w n i w i 2,R ) φ n i φ i 2,R ) λ i+ ( )) λ i+ 2,L wi n w i+ 2,L, )) φ n i φ i+ 2,L. 2,L ( (u n ) = ν(u n i ) ν, t = t ν /ν, α = α ν /ν. λ R = λ L = λ, λ u + p (ρ) = O(). CFL restriction: t ν x max ( ) λi+/2 i Z 2, notation: t = C CFLν x. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 27/39

28 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property (ρu) n+ i Scheme on ρu: ( = (ρu) n i e α t/ν2 (ν 2 (ρu 2 ) n i+ 2α x + p(ρn i+ )) (ν2 (ρu 2 ) n i ( ) + p(ρ n i )) χ x {ρ x φ} n i /2 +{ρ xφ} n i+/2 ) + α x 2 ((ρu)n i+ + 2(ρu)n i +(ρu) n i ) + λc CFL 2 ( (ρu) n i+ 2(ρu) n i +(ρu) n ) i. Formal Chapman-Enskog expansion of the approximate solution: (ρu) n i = (ρ u ) n i + O(ν). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 28/39

29 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Limit ν : (ρ u ) n i = ( p(ρn i+ ) p(ρn i ) + χ ) α 2 x 2 ({ρ xφ} n i /2 +{ρ xφ} n i+/2 ) + ( λc CFL ) ((ρ u ) n i (ρ u ) n i +(ρ u ) n ) i }{{} + ( (ρ u ) n i (ρ u ) n+ ) i, }{{} =O( t) Consistent with the analytical limit =O( x 2 ) ρ u = α xp(ρ )+ χ α ρ x φ. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 29/39

30 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Equation on ρ at the limit ν : =(ρ ) n i t e ( (ρ ) n ) i x (ρ ) n i (ρ ) n+ i ( l i /2 ( (ρ u ) n i (ρ u ) n i +l i+/2 ((ρ u ) n i+ (ρ u ) n i ( ( +λc CFL (l i+/2 e (ρ ) n ( i+) e (ρ ) n )) i ( ( l i /2 e (ρ ) n ) ( i e (ρ ) n i )) ) }{{} =O( x 2 ) ) +λc CFL χ (l i+/2 (φ n i+ φn i ) l i /2(φ n i φ n i ), }{{} =O( x) where we have set l i+/2 = e ( ) (ρ ) n i /(ρ ) n i + e ( ) (ρ ) n i+ /(ρ ) n. i+ ) )) Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 3/39

31 Construction of the approx. Riemann solver Full numerical scheme W-B property AP property Consistent with the analytical limit ( χ t ρ + x α ρ x φ ) α xp(ρ ) =. Equation on φ at the limit ν : (φ ) n+ i =(φ ) n i + D ( x φ ) n i+ ( xφ ) n i b 2 x + a (ρ ) n i + 2(ρ ) n i +(ρ ) n i+. b 4 (φ ) n i + 2(φ ) n i +(φ ) n i+ 4 Consistent with the analytical limit D xx φ = bφ aρ. The scheme is AP. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 3/39

32 Testcase Testcase 2 Testcase : perturbation of an equilibrium solution Exact equilibrium solution,2 : φ(x) = 2εbK cos( τx) τχd cos( τx) ak ( 2εK χ τd, ) b cosh D (x L) ( b cosh for x [,x], D (x L) ), for x ]x,l], { χ 2ε ρ(x) = φ(x)+ D Mτ 3/2 b tan( τx), for x [,x], τx, for x ]x,l], u(x) =, where τ = ( aχ D 2ε b) and x s.t. Natalini, Ribot, Twarogowska, CMS Twarogowska, PhD Thesis 2. b τd tan( τx) = tanh ( b D (x L) ). Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 32/39

33 φ φ Problem and objectives Testcase Testcase 2 ρ and φ with a = b = D = ε =, γ = 2, χ = 5, L =, x = T= Steady state ρ 3 ρ x.6 T= x.6 Steady state x x Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 33/39

34 Testcase Testcase 2 Testcase 2: influence of parameters on the steady state Initial conditions: ρ(x,) = +sin ( 4π x L 4 ), u(x,) =, φ(x,) =. Study Density ρ as a function of x at steady state. Influence of L and χ with a = b = D = ε =. Influence of γ with a = 2, b =, D =., ε =. Validation? No analytical solution. But results similar to those of Twarogowska,2. Natalini, Ribot, Twarogowska, CMS Twarogowska, PhD Thesis 2. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 34/39

35 Testcase Testcase Influence of L at γ = 2, χ = 3, x =...7 L=, χ=3.6 L=5, χ=3.5 ρ.8 ρ x.7 L=7, χ= x.3 L=3, χ= ρ ρ x x Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 35/39

36 Testcase Testcase 2.7 Influence of χ at γ = 2, L = 7, x =.. L=7, χ=3.8 L=7, χ= ρ ρ x 3.5 L=7, χ= x 8 L=7, χ= ρ ρ x x Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 36/39

37 Testcase Testcase 2 3 Influence of γ at χ =, L = 3, x =.. γ=2 8 γ= ρ 5 ρ x 3.5 γ= x.8 γ= ρ ρ x x Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 37/39

38 Conclusions... Problem and objectives Testcase Testcase 2 HLL consistent Riemann solver. Equilibrium states exactly preserved when γ = 2 and well approached when γ > 2. Positivity of ρ and φ. No problem with vacuum ρ =. AP scheme in the case α >.... and perspectives Understand the behaviour of the asymptotic-parabolic model. Extension of the model to the 2D. Application of this technique to a kinetic system. Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 38/39

39 References Problem and objectives Testcase Testcase 2 [] R. Natalini, M. Ribot and M. Twarogowska, A Well-Balanced Numerical Scheme for a One Dimensional Quasilinear Hyperbolic Model of Chemotaxis, Commun. Math. Sci. 2 (), pp. 3-38, 24. [2] M. Twarogowska, Numerical Approximation and Analysis of Mathematical Models Arising in Cells Movement, PhD Thesis, Università degli studi dell Aquila, 2. [3] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. on Sci. Comp. 25 (6), pp , 24. [4] A. Harten, P.D. Lax, and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM review 25, pp. 35-6, 983. [5] C. Berthon, C. Chalons, A Fully Well-Balanced, Positive and Entropy-satisfying Godunov-type method for the shallow-water equations, to appear in Math. of Comp., HAL: hal [6] E. Audusse, C. Chalons, and P. Ung, A very simple well-balanced positive and entropy-satisfying scheme for the shallow-water equations, Commun. Math. Sci. 3 (5), pp , 25. Thank you for your attention! Nice, 8 juin 25 Schéma W-B pour un modèle de chimiotactisme 39/39