Hydrodynamic limit and numerical solutions in a system of self-pr

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1 Hydrodynamic limit and numerical solutions in a system of self-propelled particles Institut de Mathématiques de Toulouse Joint work with: P.Degond, G.Dimarco, N.Wang

2 1 2 The microscopic model The macroscopic model 3

3 The classical model with 3 zones Ref: Aoki (1982), Reynolds (1986), Couzin(2002)...

4 The classical model with 3 zones Ref: Vicsek (1995), Chaté et al. (2008a),Bertin et al.(2009) Ginelli et al.(2010)...

5 The Vicsek model Discrete system X n+1 i X n i = ω n i (1) n t ω n+1 i = Ω n i + ε (2) X j X i <R ωn j ith Ω n i = ε is the noise X j X i <R ωn j, The continuous model (P.Degond, S.Motsch 08) dx i (t) = ω i (3) dt dω i (t) = (Id ω i ω i )(ν Ω i dt + 2DdB t )(4)

6 The Vicsek model Discrete system X n+1 i X n i = ω n i (1) n t ω n+1 i = Ω n i + ε (2) X j X i <R ωn j ith Ω n i = ε is the noise X j X i <R ωn j, The continuous model (P.Degond, S.Motsch 08) dx i (t) = ω i (3) dt dω i (t) = (Id ω i ω i )(ν Ω i dt + 2DdB t )(4)

7 The individual based model The microscopic model The macroscopic model Ω i = dx i (t) dt = v i (5) v i (t) = v 0 ω i + µf i (6) dω i (t) = (Id ω i ω i )(ν Ω i dt K( Xi X j )ω j K( X i X j )ω j + 2DdB t + αv i dt) (7) F i = 1 N N j=1 φ(x i X j ) The repulsive force acts on the i-th particle X_k R w_k v_k

8 The individual based model The microscopic model The macroscopic model Ω i = dx i (t) dt = v i (5) v i (t) = v 0 ω i + µf i (6) dω i (t) = (Id ω i ω i )(ν Ω i dt K( Xi X j )ω j K( X i X j )ω j + 2DdB t + αv i dt) (7) F i = 1 N N j=1 φ(x i X j ) The repulsive force acts on the i-th particle X_k R w_k v_k

9 The microscopic model The macroscopic model Mean field limit The density of particles f (x, ω, t) satisfies: t f + x (vf ) + ω (Gf ) + ω (Hf ) D ω f = 0 (8) where v = v 0 ω µ φ( x y )f (y, ω, t)dω (9) G(x, ω, t) = ν(id ω ω) Ω f (x, t) (10) Ω f (x, t) = j f j f (11) j f (x, t) = K( x y )f (y, ω, t)dydω (12) H(x, ω, t) = α(id ω ω)v (13)

10 Scaling The microscopic model The macroscopic model We define the regime we are interested in: The interaction regimes of kernels K, φ are small. The diffusion and social terms are large. The alignment term prevails over the repulsion term The other terms are kept order of 1.

11 Scaling The microscopic model The macroscopic model We define the regime we are interested in: The interaction regimes of kernels K, φ are small. The diffusion and social terms are large. The alignment term prevails over the repulsion term The other terms are kept order of 1.

12 Scaling The microscopic model The macroscopic model We define the regime we are interested in: The interaction regimes of kernels K, φ are small. The diffusion and social terms are large. The alignment term prevails over the repulsion term The other terms are kept order of 1.

13 Scaling The microscopic model The macroscopic model We define the regime we are interested in: The interaction regimes of kernels K, φ are small. The diffusion and social terms are large. The alignment term prevails over the repulsion term The other terms are kept order of 1.

14 The microscopic model The macroscopic model Finally, the function f ε (x, ω, t) satisfies ε( t f ε + x (vf ε ) + ω (Hf ε ) + ω (L ε f ε )) = Q(f ε ) (14) with v(x, ω, t) = v 0 ω µφ ( f ε dω) (15) G ε (x, ω, t) = ν(id ω ω) Jε f (x, t) Jf ε(x, t) (16) L ε f ε = k 0ν(Id ω ω)lf ε ε(x, t) (17) lf ε ε(x, t) = (Id Ω Ω) (Jε f (x, t)) Jf ε (x, t) (18) Jf ε (x, t) = f ε ωdω (19)

15 The microscopic model The macroscopic model The Properties of Q(f ) Q(f ) = (Gf ) + D f (20) The equilibrium of Q(f ) M Ω (ω) = Cexp( ω Ω d ) (21) with d = D/ν, for Ω is an arbitrary direction.

16 The macroscopic model The microscopic model The macroscopic model The density ρ(x, t) and the average direction Ω(x, t) satisfy the following equations t ρ + x (ρu 1 ) = 0 (22) t Ω + ρ(u 2 )Ω + δ(id Ω Ω) x ρ = γ(id Ω Ω) (ρω) (23) with u 1 = c 1 v 0 Ω µφ ρ, u 2 = c 2 v 0 Ω µφ ρ, δ = v 0 d + αµφρ(2d + c 2 ), γ = (2d + c 2 )k 0

17 We want to numerically solve the macroscopic model t ρ + x (ρu 1 ) = 0 t Ω + ρ(u 2 )Ω + δ(id Ω Ω) x ρ = γ(id Ω Ω) (ρω) Ω = 1 The model is non- conservative has a geometric constraint = We replace the geometric constraint by a relaxation operator. (Ref: S.Motsch, L.Navoret)

18 We want to numerically solve the macroscopic model t ρ + x (ρu 1 ) = 0 t Ω + ρ(u 2 )Ω + δ(id Ω Ω) x ρ = γ(id Ω Ω) (ρω) Ω = 1 The model is non- conservative has a geometric constraint = We replace the geometric constraint by a relaxation operator. (Ref: S.Motsch, L.Navoret)

19 We want to numerically solve the macroscopic model t ρ + x (ρu 1 ) = 0 t Ω + ρ(u 2 )Ω + δ(id Ω Ω) x ρ = γ(id Ω Ω) (ρω) Ω = 1 The model is non- conservative has a geometric constraint = We replace the geometric constraint by a relaxation operator. (Ref: S.Motsch, L.Navoret)

20 We want to numerically solve the macroscopic model t ρ + x (ρu 1 ) = 0 t (ρω) + (ρu 2 Ω) + δ ρ γ (ρω) = ρ η (1 Ω 2 ) Ω 2 The model is non- conservative has a geometric constraint = We replace the geometric constraint by a relaxation operator. (Ref: S.Motsch, L.Navoret)

21 Splitting method The idea is to split the relaxation model into two parts. Firstly, we consider this part t ρ + (ρu 1 ) = 0 (24) t (ρω) + (ρu 2 Ω) + δ ρ γ (ρω) = 0 (25) and the relaxation part t ρ = 0 t (ρω) = ρ η (1 Ω 2 ) Ω 2. (26)

22 Splitting method The idea is to split the relaxation model into two parts. Firstly, we consider this part t ρ + (ρu 1 ) = 0 (24) t (ρω) + (ρu 2 Ω) + δ ρ γ (ρω) = 0 (25) and the relaxation part t ρ = 0 t (ρω) = ρ η (1 Ω 2 ) Ω 2. (26)

23 The convergence of the splitting scheme Problem: The density is uniform on the domain The orientation is a vortex

24 Reference P. Degond and S. Motsch Mathematical Models and Methods in Applied Sciences. Vol.18, Suppl.(2008) Juan A. Henkes, Y. Fily, M.C. Marchetti, Physical review 84, (2011). B.Szabo, G.J.Szollozi, B. Gonci, Zs. Juranyi, D.Selmeczi, T. Vicsek, Physical review E (2006). A. B.T Barbaro, P. Degond (2010) preprint. T.Vicsek, A. Czirok, E. Ben - Jacob, I. Cohen and O. Shochet, Phys. Rev. Lett. 75(1995)

25 Thank you for listening!