Finite volume methods for dissipative problems Claire - PowerPoint PPT Presentation

Finite volume methods for dissipative problems Claire Chainais-Hillairet CEMRACS summer school Marseille, July 2019, 15-20

Lecture 2 : Dissipative problems and long time behavior

Overview Evolutive equation t → ∞ Steady-state or system of equations F ∞ F ∆ x, ∆ t → 0 ∆ x → 0 Scheme for the Scheme evolutive problem for the equilibrium F ∞ F ∆ x, ∆ t ∆ x

Overview Evolutive equation t → ∞ Steady-state or system of equations F ∞ F ∆ x, ∆ t → 0 ∆ x → 0 n → ∞ Scheme for the Scheme evolutive problem for the equilibrium F ∞ F ∆ x, ∆ t ∆ x

Framework : dissipative problems � � ∂ t u + Au = 0 , t ≥ 0 , u ∞ , ( F ∞ ) ( F ) u (0) = u 0 . Au ∞ = 0 . Main features Existence of a Lyapunov convex function E satisfying d dtE ( u ) = −� Au, E ′ ( u ) � ≤ 0 . E is general given by the physics : it is a physical energy or entropy, which is dissipated along time. D ( u ) = � Au, E ′ ( u ) � is the dissipation of energy/entropy. The steady-state is a minimizer for E .

Dissipativity and long time behavior � � ∂ t u + Au = 0 , t ≥ 0 , u ∞ , ( F ∞ ) ( F ) u (0) = u 0 . Au ∞ = 0 . Main features d dtE ( u ) = − D ( u ) D ( u ) ≥ λE ( u ) = ⇒ exponential decay : E ( u ) ≤ E ( u 0 ) e − λt . D ( u ) ≥ KE ( u ) 1+ γ = ⇒ polynomial decay ∼ t − 1 /γ

Overview Evolutive equation t → ∞ Steady-state or system of equations F ∞ F upwind centered SG ∆ x, ∆ t → 0 ∆ x → 0 + exponential convergence ? n → ∞ Scheme for the Scheme evolutive problem for the equilibrium F ∞ F ∆ x, ∆ t ∆ x

Outline of the lecture Some example of dissipative problems 1 Long time behavior of the Fokker-Planck equation 2 Long time behavior of the porous media equation 3

Shallow water equations with viscous terms Incoming fluxes h : water height Lake b : ground topography Sea level h b u : velocity Sediments Shallow water equations with friction � ∂ t h + div( h u ) = 0 ∂ t h u + div( h u ⊗ u ) + gh ∇ ( h + b ) = − C | u | u E ( h, u ) = 1 � � h | u | 2 + g ( h + b ) 2 � Energy/dissipation 2 T 2 � T 2 C | u | 3 D ( h, u ) =

Study over large time scales ❑ Peton, ’18 � ∂ t h + div( h u ) = 0 gh ∇ ( h + b ) = − C | u | u ⇒ u = − ( g C ) 1 / 2 h 1 / 2 |∇ ( h + b ) | − 1 / 2 ∇ ( h + b ) = Conservation law for the water height ∂ t h + div( − Kh 3 / 2 |∇ ( h + b ) | − 1 / 2 ∇ ( h + b )) = 0 degenerate 3/2-laplacian equation, with drift.

Dissipative behavior � ∂ t h + div( − Kh 3 / 2 |∇ ( h + b ) | − 1 / 2 ∇ ( h + b )) = 0 + no-flux boundary condition E ( h ) = 1 � ( h + b ) 2 Energy/dissipation 2 Ω d � dtE ( h ) = ∂ t h ( h + b ) Ω � − Kh 3 / 2 |∇ ( h + b ) | − 1 / 2 ∇ ( h + b ) � � = − div ( h + b ) Ω � Kh 3 / 2 |∇ ( h + b ) | 3 / 2 = − Ω Steady states h = 0 or h + b = constant.

Saltwater intrusion model Management of fresh water resources in costal regions Quantities of interest height of the freshwater height of the interface with the saltwater

Saltwater intrusion model ❑ Escher, Laurenc ¸ot, Matioc, ’12 , ❑ Laurenc ¸ot, Matioc, ’14, ’17 ❑ Jazar, Monneau, ’14 Notations Assumptions z = b + g + f Thin layers, sharp interfaces dry sol z = b + g Large time scales freshwater z = b Incompressible immiscible phases saltwater bedrock Mainly horizontal displacements ρ : ratio of the densities, ρ = ρ f < 1 , ρ s ν : ratio of the kinematic viscosities, ν = ν s ∈ (0 , + ∞ ) ν f

Saltwater intrusion model  ∂ t f + div( − νf ∇ ( f + g + b )) = 0 ,   ∂ t g + div( − g ∇ ( ρf + g + b )) = 0 ,  + no-flux boundary conditions  Dissipative behavior ρ 2( f + g ) 2 + 1 − ρ � g 2 + b ( ρf + g ) . E ( f, g ) = 2 Ω d � dtE ( f, g ) = ∂ t f ( ρ ( f + g + b )) + ∂ t g ( ρf + g + b ) Ω � � 2 + g � 2 � � � � = − � ∇ ( ρ ( f + g + b )) � ∇ ( ρf + g + b ) ρνf Ω � � 2 + g � 2 � � � � = − ρνf � ∇ Φ f � ∇ Φ g Ω

About porous media equation : Ω = R d , b = 0 , f = 0 ∂ t g + div( − g ∇ g ) = 0 Self-similar solutions Passage to self-similar variables : x d (1 + t ) 1 / ( d +2) , g ( t, x ) = e − τ d +2 u ( τ, ξ ) τ = log(1 + t ) , ξ = Nonlinear Fokker-Planck equation on u : | x | 2 ∂ t u + div( − u ∇ ( u + 2( d + 2))) = 0 Self-similar solution in g ⇐ ⇒ steady-state in u Exponential decay in u and polynomial decay in g ❑ Barenblatt, ’1952 ❑ Carrillo, Toscani, ’00

Salt water intrusion model : self-similar solutions Initial system with b = 0 , Ω = R 2 � ∂ t f + div( − νf ∇ ( f + g )) = 0 , ∂ t g + div( − g ∇ ( ρf + g )) = 0 , Self-similar variables and new system � � 1 x (1 + t ) 1 / 2 ( ˜ ( f, g )( t, x ) = f, ˜ g ) log(1 + t ) , (1 + t ) 1 / 4 ∂ t f + div( − νf ∇ ( f + g + b  ν )) = 0 ,      ∂ t g + div( − g ∇ ( ρf + g + b )) = 0 ,  b ( x ) = | x | 2     8

Salt water intrusion model : self-similar solutions ρ 2( f + g ) 2 + 1 − ρ g 2 + b ( ρ � E ( f, g ) = ν f + g ) , 2 Ω � � 2 + g � 2 � � � � D ( f, g ) = ρνf � ∇ Φ f � ∇ Φ g Ω with Φ f = f + g + b ν , Φ g = ρf + g + b Characterization of the steady-states (self-similar solutions) Stationary solutions have vanishing fluxes : F ∇ Φ F = 0 , G ∇ Φ G = 0 . The minimizer of the energy is a stationary solution. There exists a unique minimizer of E which is radially symmetric. ❑ Ait Hammou Oulhaj, Canc` es, C.-H., Laurenc ¸ot, ’19 ❑ Laurenc ¸ot, Matioc, ’14

Self-similar profiles G F F F, G F, G ν ρ F 1 G 0 ν ⋆ ν ⋆ ν ⋆ 1 3 2 G F, G F, G ρ 2 M f ρ M f M g + 1 3 = 1 + (1 − ρ ) M f M g ν ⋆ ν ⋆ ν ⋆ 1 = , 2 = , . 1 + ρ ( M f M f M g M g − 1) M g + 1

Numerical experiments Topological change of E G near the critical value ν ⋆ 1 = 0 . 81 ν = ν ⋆ ν = 0 . 80 1 = 0 . 81 ν = 0 . 82

Numerical experiments Topological change of E F near the critical value ν ⋆ 3 = 1 . 1 ν = ν ⋆ ν = 1 . 09 3 = 1 . 10 ν = 1 . 11

Convergence towards the steady-state ( f ( t ) , g ( t )) → ( F, G ) in L 2 ( R 2 ; R 2 ) as t → ∞ 10 -4 10 -4 10 -5 10 -5 10 -6 Cexp( -0.0310t) Cexp(-0.0992t) 10 -6 10 -7 0 5 10 15 20 0 5 10 15 20 ν = 0 . 4 ν = 0 . 9 10 -4 10 -4 10 -5 10 -5 Cexp(-0.0802t) 10 -6 10 -6 Cexp(-0.0344t) 10 -7 10 -7 0 5 10 15 20 0 5 10 15 20 ν = 0 . 95 ν = 2

Exponential convergence towards the steady-state ? Rate of convergence with respect to ν 0.42 0.39 0.36 0.33 0.3 0.27 0.24 0.21 p 0.18 0.15 0.12 0.09 0.06 0.03 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ν

Outline of the lecture Some example of dissipative problems 1 Long time behavior of the Fokker-Planck equation 2 Long time behavior of the porous media equation 3

Focus on Fokker-Planck equations  ∂ t f + ∇ · J = 0 , J = −∇ f + U f, in Ω × R +    J · n = 0 on Γ N × R +    f = f D on Γ D × R +      f ( · , 0) = f 0 > 0 .  Some references ❑ Carrillo, Toscani, ’98 ❑ Arnold, Markowich, Toscani, Unterreiter, ’01 ❑ Carrillo et al., ’01 ❑ Bodineau, Lebowitz, Mouhot, Villani, ’14 ❑ Gajewski, Gr¨ oger, ’86, ’89 ❑ J¨ ungel, ’95

Thermal equilibrium, when U = −∇ Ψ  ∂ t f + ∇ · J = 0 , J = −∇ f −∇ Ψ f, in Ω × R +   J · n = 0 on Γ N × R +    f = f D on Γ D × R +     f ( · , 0) = f 0 > 0 .  f = λe − Ψ = ⇒ J = 0 f ∞ = λe − Ψ Existence of a thermal equilibrium � � if Γ D = ∅ , with λ = e − Ψ , f 0 / Ω Ω if log f D + Ψ D = α , with λ = e α . J = − f ∇ (log f + Ψ) = − f ∇ log f = ⇒ f ∞

Entropy-dissipation property J = − f ∇ log f ∂ t f + ∇ · J = 0 , f ∞ Relative entropy Φ 1 ( x ) = x log x − x + 1 � f ∞ Φ 1 ( f H 1 ( t ) = f ∞ ) Ω Dissipation of the entropy d dtH 1 ( t ) = − D 1 ( t ) , 2 � ∇ log f � � � � � ≥ 0 with D 1 ( t ) = f � � f ∞ � Ω

Exponential decay towards thermal equilibrium No-flux boundary conditions � � � f ∞ conservation of mass : f = f 0 = Ω Ω Ω � f log( f/f ∞ ) H 1 ( t ) = Ω 2 � � f |∇ log( f/f ∞ ) | 2 = 4 f ∞ � � � � ∇ D 1 ( t ) = f/f ∞ � � � Ω Ω thanks to Logarithmic Sobolev inequality : 0 ≤ H 1 ( t ) ≤ H 1 (0) e − κt and with Csiszar-Kullback inequality : � f ( t ) − f ∞ � 2 1 ≤ 2 H 1 (0) e − κt

Exponential decay towards thermal equilibrium Dirichlet boundary conditions Upper and lower bounds on f and f ∞ � Φ 1 ( f ) − Φ 1 ( f ∞ ) − ( f − f ∞ )Φ ′ 1 ( f ∞ ) H 1 ( t ) = Ω c � f ( t ) − f ∞ � 2 2 ≤ H 1 ( t ) ≤ C � f ( t ) − f ∞ � 2 2 � f |∇ (log f − log f ∞ ) | 2 D 1 ( t ) = Ω with Poincar´ e inequality : D 1 ( t ) ≥ C� f ( t ) − f ∞ � 2 2 Conclusion : c � f ( t ) − f ∞ � 2 2 ≤ H 1 ( t ) ≤ H 1 (0) e − κt