An asymptotic preserving and well-balanced scheme for a chemotaxis model Christophe Berthon jointed work with Marianne Bessemoulin, University of Nantes (France) Anaïs Crestetto, University of Nantes (France) Françoise Foucher, University of Nantes (France) Hélène Mathis, University of Nantes (France) Madison, May 25
Main motivations Hyperbolic systems of conservation law with source term t w + x f(w) = ν S(w) w Ω Steady states x f(w) = ν S(w) Manifold given by Asymptotic diffusive regime: ν Finite volume schemes Numerical approximations of the weak solutions Robustness Exact capture of (a part of) M Asymptotic preserving M = {w Ω; g(w) = }
Outline A chemotaxis model (with A. Crestetto and F. Foucher) Godunov type well-balanced scheme Model and main properties Godunov type scheme Characterization of the approximate Riemann solver Robustness and well-balanced properties Asymptotic preserving property Asymptotic convergence rate (with M. Bessemoulin and H. Mathis) relative entropies to estimate the asymptotic convergence rate A continuous estimation by C. Lattanzio and A. Tzavaras Extension to a semi-discrete scheme
A model of Chemotaxis (Ribot et al 22-24) t ρ + x ρu = t ρu + x (ρu 2 + p(ρ)) = χρ x Φ αρu t Φ D xx Φ = aρ bφ Steady states at rest u = e(ρ) χφ = cste solutions of D xx Φ bφ = aρ if ρ = : if ρ >, C < : if ρ >, C > : e(ρ) = δ χ, α, D, a, b given parameters Ω = {w R 3 ; ρ, u R, Φ } p(ρ) = δρ γ γ γ ργ + cste ( ϕ (x) = A cosh x ) ( b + B sinh x ) b, ( ϕ (x) = A cos x ) ( C + B sin x ) C ϕ p, ρ (x) = χ (ϕ (x) K), ( 2δ ϕ (x) = A cosh x ) ( C + B sinh x ) C ϕ p, ρ (x) = χ (ϕ (x) K), 2δ where A and B are some constants, C = D ( ) b aχ 2δ and ϕp = Kaχ 2δb aχ.
Asymptotic behavior Rescaling: t t/ν (long time) and α α/ν (dominant friction) Limit ν to get a diffusive regime { t ρ = x ( x p χρ x Φ) D xx Φ = bφ aρ Objectives Robustness (ρ and Φ ) Steady state preserving (well-balanced) Asymptotic preserving Godunov type strategy (CB and Chalons 25)
Godunov type scheme { t ρ + x ρu = t ρu + x (ρu 2 + p(ρ)) = χρ x Φ αρu Φ given Approximate Riemann solver: w λ L λ R w L w L w R w R λ L < < λ R HLL type solver Source term stationary contact wave x Harten-Lax-van Leer consistency condition /2 /2 w (x, t; w L, w R ) dx = /2 /2 w R (x, t; w L, w R ) dx /2 /2 w (x, t; w L, w R ) dx = 2 (w L+w R )+ t (λ L(w L w L)+λ R (w R w R ))
Because of the source term, w R stays unknown w L (t) w L t w R (x, t; w L, w R ) w R w R (t) x constant is not a natural solution /2 /2 t ( ) t w + x f(w) = S(w) dxdt /2 /2 Approximation w R (x, t; w L, w R ) dx = 2 (w L + w R ) + t f(w R (/2, t))dt + w R ( /2, t) w L w R (/2, t) w R t /2 /2 t f(w R ( /2, t))dt S(w)dxdt
As a consequence /2 /2 /2 /2 S R = t ρ R (x, t; w L, w R ) dx 2 (ρ L + ρ R ) t (ρ Ru R ρ L u L ) (ρu) R (x, t; w L, w R ) dx 2 (ρ Lu L + ρ R u R ) t (ρ Ru 2 R + p R ρ L u 2 L p L ) + ts R α t /2 /2 t /2 /2 χρ R x Φdxdt (ρu) R (x, t; w L, w R ) dxdt Approximation S R S to be defined (independently from t) Approximation /2 /2 solution of the following integral equation (ρu) R (x, t; w L, w R ) dx F( t) F( t) = 2 (ρ Lu L +ρ R u R ) t (ρ Ru 2 R +p R ρ L u 2 L p L )+ ts α t F(t)dt
Consistency conditions 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 Lu L) + λ R (ρ Ru R ρ R u R ) = α t ( e α t ) ( α 2 (ρ Lu L + ρ R u R ) (ρ R u 2 R + p R ρ L u 2 L p L ) + S ) Flux continuity: ρ L u L = ρ R u R Well-balanced conditions S = χ p R p L e R e L (Φ R Φ L ) e L ρ L ρ L χϕ L = e R ρ R ρ R χϕ R
Approximate Riemann solver: w λ L λ R Φ L Φ R Φ L Φ R x { t Φ + x Ψ = aρ bφ Ψ = x Φ ρ given Harten-Lax-van Leer consistency condition /2 /2 Φ (x, t; w L, w R ) dx = /2 /2 Φ R (x, t; w L, w R ) dx Main difficulty: Evaluate Φ R /2 t /2 ( t Φ D x Ψ) dxdt = a /2 /2 t ρ R (x, t)dxdt b Φ R (x, t)dxdt Approximations /2 /2 /2 /2 ρ R (x, t)dx 2 (ρ L + ρ R ) t x Ψ(x, t)dx t (Ψ R Ψ L ) t (ρ Ru R ρ L u L )
Approximate Riemann solver: w Godunov type scheme λ L λ R Φ L Φ R Φ L Φ R x { t Φ + x Ψ = aρ bφ Ψ = x Φ ρ given Harten-Lax-van Leer consistency condition /2 /2 Φ R (x, t)dx G( t) G( t) solution of the following integral equation G( t) = 2 (Φ L + Φ R ) + D t ( ) t (Ψ R Ψ L ) + a 2 (ρ L + ρ R ) t2 2 (ρ Ru R ρ L u L ) b t G(t)dt
Godunov type scheme ( w n+ i Φ n+ i = w n i t = Φ n i t Definition of Ψ n i We impose ( ) ( )) λ i 2,R wi n w i 2,R λ i+ 2,L wi n ( ( ) ( w i+ 2,L )) λ i 2,R Φ n i Φ i 2,R λ i+ 2,L Φ n i Φ i+ 2,L Ψ n i = 2 (Φn i+ Φ n i ) E() E() is consitent with lim E() = E() = 2 b ( 2 b ) cos 2 C ( C ) 2 cos if ρ = if ρ >, C < 2 2 C ( C ) cosh if ρ >, C > to recover the steady states
Theorem Robustness (adopting a local cut-off, Chalons et al 24) ρ L,R = min(max(, ρ L,R), 2ρ HLL ) Φ L,R = min(max(, Φ L,R), 2Φ HLL ) ρ n+ i and Φ n+ i Well-balance property
Asymptotic diffusive regime Rescalling u = νu ν, t = t ν /ν and α = α ν /ν, Rescalled system given by ν t ν ρ 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 ν ϕ 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 ( χ t ν ρ + x α ν ρ x ϕ ) α ν xp(ρ ) D xx ϕ = bϕ aρ, ρ u = α ν xp(ρ ) + χ α ν ρ x ϕ. =,
w n+ i Φ n+ i Rescalling = w n i t = Φ n i t Asymptotic preserving scheme ( ( ) λ i 2,R wi n w i 2,R λ i+ ( )) 2,L wi n ( ( ) ( w i+ 2,L )) λ i 2,R Φ n i Φ i 2,R λ i+ 2,L Φ n i Φ i+ 2,L (u n ) = ν(u n i ) ν, t = t ν /ν and α = α ν /ν, Simplification λ R = λ L = λ λ u + p (ρ) = O() CFL restriction t ν max ( ) λi+/2 2,
In the limite ν ρ n+ i (ρu) n i =ρ n i t e (ρ n i ) ρ n i ( ( l i /2 (ρu) n i (ρu) n ) ( i + li+/2 (ρu) n i+ (ρu) n i )) ) ( ( ) + λc CF L (l i+/2 e ρ n i+ e (ρ n i ) ) ( l i+/2 e (ρ n i ) e ( ρ n i ( ) + λc CF L χ l i+/2 (ϕ n i+ ϕ n i ) l i /2 (ϕ n i ϕ n i ) = ( p(ρn i+ ) p(ρn i ) + χ ) α 2 2 ({ρ xϕ} n i /2 + {ρ xϕ} n i /2 ) + ( λc CF L ) ((ρu) n i+ 2(ρu) n i + (ρu) n i 2 2 )) ) ( + (ρu) n i (ρu) n+ ) i D ( xϕ) n i+ ( xϕ) n i 2 where we have set =b ϕn i + 2ϕn i + ϕn i+ 4 l i+/2 = a ρn i + 2ρn i + ρn i+ 4 e (ρ n i ) /ρn i + e ( ρ n i+) /ρ n i+ + (ϕ n+ i ϕ n i ) The scheme is consitent with the asymptotic regime
Numerical results Influence of γ at χ =, L = 3, =.. 3 =2 8 =3 7 25 6 2 5 5 4 3 2 5.5.5 2 2.5 3 x.5.5 2 2.5 3 x 3.5 =4.8 =5 3.6.4 2.5.2 2.5.8.6.4.5.2.5.5 2 2.5 3 x.5.5 2 2.5 3 x
Convergence rate as ν Simpler model: p-system { t τ x u = t u + x p(τ) = σu Asymptotic regime Relative entropy Rescalling t t/ϵ u ϵu ϵ α α/ϵ t τ x ū = t p( τ) = σ ū { t τ ϵ x u ϵ = ϵ 2 t u ϵ + x p(τ ϵ ) = σu ϵ η ε (τ, u τ, u) = ε2 2 (u u)2 P (τ τ) with P (τ τ) = P (τ) P (τ) p(τ)(τ τ) To satisfy t η ε (τ ε, u ε τ, u)+ ε 2 xψ(τ ε, u ε τ, u) = σ(u ε u) 2 + σ xxp(τ) p(τ ε τ)+ ε2 σ xtp(τ)(u ε u)
Lattanzio and Tzavaras estimation Introduce ϕ(t) = η(τ ε, u ε τ, u)dx Assume τ c > xx p(τ) L (Q T ) K < + xt p(τ) L 2 (Q T ) K < + R Then ϕ(t) C(ϕ() + ε 4 ) t [, T ) Objective: Recover this estimation with numerical approximations
Semi-discrete scheme p-system: semi-discrete scheme (τ i (t), u i (t)) dτ i dt = 2 (u i+ u i ) + du i dt = 2ε 2 (p(τ i+) p(τ i )) σ ε 2 u i + λ 2 (τ i+ 2τ i + τ i ) asymptotic regime: semi-discrete scheme (τ i (t), u i (t)) dτ i dt = u i = σ 2 (u i+ u i ) + 2 (p(τ i+) p(τ i )) λ 2 (u i+ 2u i + u i ) λ 2 (τ i+ 2τ i + τ i )
Relative entropy: η i = η(τ i, u i τ i, u i ) dη i dt + ψ i+/2 ψ i /2 ε 2 = σ(u i u i ) 2 + p(τ i+2 ) 2p(τ i ) + p(τ i 2 ) σ (2) 2 p(τ i τ i )+ ε 2 ( ) d p(τ i+ ) p(τ i ) (u i u i ) + R τ i + R u i σ dt 2 R τ i = ε 2 λ 2 (u i+ 2u i + u i )(u i u i ) R u i = λ 2 ((p(τ i) p(τ i ))(τ i+ 2τ i + τ i ) + p (τ i )(τ i+ 2τ i + τ i )(τ i τ i )) By summation: ϕ(t) = η i(t) t ϕ(t) ϕ() = σ + ε2 σ t t (u i u i ) 2 ds + σ ( ) d p(τ i+ ) p(τ i ) (u i u i )ds + dt 2 p(τ i+2 ) 2p(τ i ) + p(τ i 2 ) (2) 2 t p(τ i τ i )ds (R τ i + R u i )ds
Sketch of the proof of the convergence rate Estimations of Lattanzio and Tzavaras σ ε 2 σ t p(τ i+2 ) 2p(τ i ) + p(τ i 2 ) (2) 2 p(τ i τ i )ds C σ D xxp(τ) ( ) d p(τ i+ ) p(τ i ) (u i u i )ds dt 2 t σ 2 t t ϕ(s)ds (u i u i ) 2 ds + C D xt p(τ) 2 ε 4 Estimation of the viscous terms t R u i ds λθ t (u i u i ) 2 ds + C(θ, D xx u 2 )ε 4 2 t t R τ i ds C( D x τ, D xx τ ) ϕ(s)ds
Theorem: Assume τ c > and ( T ( ) D tx p(τ) L 2 := d p(τ i+ ) p(τ i ) 2 /2 ds) K dt 2 D xx p(τ) L := sup sup p(τ i+2 ) 2p(τ i ) + p(τ i 2 ) t [,T ] (2) 2 K ( T ( ) ) 2 /2 τ i+ 2τ i + τ i D xx τ L := 2 ds K Then D x τ L := sup sup t [,T ] ( T D xx u L 2 := to get the convergence rate τ i+ τ i K ( ) 2 ui+ 2u i + u i 2 ds ϕ(t) ϕ() + Cε 4 + C t ϕ(s)ds ) /2 K ϕ(t) ( ϕ() + Cε 4) e CT t T
Jin-Pareschi-Toscani scheme (98) Refomulation t τ x u = Splitting scheme τ n+ i = τ n+ 2 i u n+ i u n+ 2 i t = ε 2 t u + x p(τ) = ( σ u + ( ε 2 ε 2 ) x p(τ) ) Expected result τ n+ 2 i = τ n i t u n+ 2 i = u n i t σ u n+ i + ( ε 2 ) ( G τ i+ 2 ( G u i+ 2 p(τ n+ i+ 2 ϕ n = η(τ n i, u n i τ n i, u n i ) ) G τ i 2 ) G u i 2 ) p(τ n+ i 2 ) τ n+ i+ 2 = ( τ n+ i + τ n+ ) i+ 2 ( ϕ + Cε 4) e CN n N
Numerical experiments.. Phi(eps) eps^4.. Phi(eps) eps^4 e-6 e-6 e-8 e-8 e- e- e-2 e-2 e-4 e-4 eps 2 si x < τ (x) = si x > e-6.... u (x) = δ e-6.... eps τ (x) = exp( x 2 ) + u (x) = x τ (x) Number of cells: 8 Final time of simulations: T = 2
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