Discrete entropy methods for nonlinear diffusive equations

Transcription

1 Discrete entropy methods for nonlinear diffusive equations 1 Ansgar Jüngel Vienna University of Technology, Austria Joint with C. Chainais-Hillairet (Lille), J. A. Carrillo (Imperial), E. Emmrich (TU Berlin), M. Bukal (Zagreb), J.-P. Milišić (Zagreb), S. Schuchnigg (Vienna) Introduction: Entropy methods Implicit Euler finite volumes Higher-order time schemes Extension

2 Introduction 2 Entropy = Measure of molecular disorder or energy dispersal Entropy in physics: Introduced by Clausius (1865) in thermodynamics Boltzmann, Gibbs, Maxwell: statistical interpretation Shannon (1948): concept of information entropy Entropy in mathematics: entropy Lyapunov functional Lax (1973): entropy in hyperbolic conservation laws DiPerna Lions (1989): entropy in kinetic theory Connections to functional inequalities (Gross 1975) and stochastic diffusion processes (Bakry-Emery 1983) Toscani (1997): kinetic Fokker-Planck equations Main aim: Derive (sharp) decay rates to equilibrium

3 Introduction 3 Example: Fokker-Planck eq. t u = div( u+xu) in R d Entropy: H[u] = Ω ulog u u dx, u = e x 2 /2 : steady state Entropy-dissipation inequality: D[u] = dh dt [u] = R d u log u u 2 dx 0 Second-order time derivative: (Bakry-Emery 1983) d 2 H dt [u]= ( 2u log u 2 ) 2 R d u +(...) 2 dx 2 dh dt dh dt 2H Consequence 1 : Exponential decay to equilibrium H[u(t)] H[u(0)]e 2t, t 0 Consequence 2 : Logarithmic Sobolev inequality: H[u] = ulog u u dx 1 Ω 2 T D[u] = 2 u 2 d u u dx Benefit: very robust, in particular for nonlinear problems

4 Introduction 4 Abstract entropy-dissipation method Setting: u solves A(u) = 0, u solves t u+a(u) = 0, t > 0, u(0) = u 0 Lyapunov functional: H[u] satisfies dh dt [u(t)] 0 for t 0 Entropy: convex Lyapunov functional H[u] such that D[u] := dh dt [u] = A(u),H [u] 0 Bakry-Emery approach: show that, for κ > 0, d 2 H dt [u] κdh 2 dt [u] D[u] = dh [u] κh[u] dt Consequences: dh dt κh implies that H[u(t)] H[u(0)]e κt t > 0 H[u] κ 1 D[u] corresponds to convex Sobolev inequality

5 Introduction 5 Setting: t u+a(u) = 0, t > 0, u(0) = u 0 Program: Develop discrete entropy methods Tasks: Entropy-dissipating discrete schemes: dh dt [u] 0 Discrete large-time asymptotics: H[u(t)] H[u(0)]e κt Discrete convex Sobolev inequality: H[u] κ 1 D[u] Discretizations: Time discretizations: Euler implicit, higher-order schemes... Space discretization: Finite volumes, finite elements, DG... Fully discrete schemes Higher-order minimizing movement schemes Equations: Porous-medium, quantum diffusion, systems...

6 Overview Previous results Discrete logarithmic Sobolev inequalities: Holley-Stroock 1987: Ising models for spin systems Zegarlinski 1990: Infinite lattice systems Talagrand 1993: Discrete cube {0,1} n Discrete Markov processes: Bonciocat-Sturm 2009: Rough curvature bounds Caputo-Dai Pra-Posta 2009: Exponential decay Chow-Huang-Li-Zhou 2012: Free energy & finite graphs Mielke 2013: Geodesic convexity of entropy Maas 2011, Erbar-Maas 2014: Discrete gradient flows Entropy-dissipating schemes: Tadmor 1987, Carrillo-A.J.-Tang 2003, Glitzky-Gärtner 2009 Focus on stable numerical schemes for nonlinear eqs. 6

7 Overview 7 Introduction Implicit Euler finite-volume scheme Time-discrete one-leg multistep scheme Time-discrete Runge-Kutta scheme Extension

8 Implicit Euler finite-volume scheme 8 Example: t u = u β with no-flux boundary conditions Continuous case: entropy H α [u] = Ω uα dx ( Ω udx)α dh α = d u α dx = α u α 1 u β dx dt dt Ω Ω = 4αβ u (α+β 1)/2 2 dx CH α [u] (α+β 1)/α α+β 1 Ω follows from Beckner inequality: (f = u (α+β 1)/2 ) ( ) pq f q dx f 1/p dx C B f q, q 2 L 2 (Ω) Ω Ω Standard Beckner inequality: q = 2 (meas(ω) = 1) Proof: DifferentiateL p interpolationinequality(dolbeault) and use generalized Poincaré-Wirtinger inequality Task: Translate computations to discrete case

9 Implicit Euler finite-volume scheme 9 Finite-volume scheme: Ω = K Control volumes K, edges σ = K L Transmissibility coeff.: τ σ = K /d σ K (u k K u k 1 K )+τ σ=k L K x K σ d σ τ σ ((u k K) β (u k L) β ) = 0 Discrete case: H d α[u] = K K uα K ( K K u K) α H d α[u k ] H d α[u k 1 ] = K K ((u k K) α (u k 1 K )α ) K (u k K) α 1 (u k K uk k 1 ) K C 1 (u k ) (α+β 1)/2 2 H 1 C 2 Hα[u d k ] (α+β 1)/α Follows from discrete Beckner inequality Proof: Use discrete Poincaré-Wirtinger inequality (Bessemoulin-Chatard, Chainais-Hillairet, Filbet 2012) x L L

10 Implicit Euler finite-volume scheme 10 Theorem: (Chainais-Hillairet/A.J./Schuchnigg, 2013) H d α[u k ] (C 1 t k +C 2 ) α/(β 1), α > 1, β > 1 H d α[u k ] H d α[u 0 ]e λt k, 1 < α 2, β > 0 First-order entropies? H α [u] = Ω uα/2 2 dx 4 Continuous case: Let (α,β) M d dh α = α div(u α/2 1 u α/2 ) u β 3 dx dt Ω 2 C u α+β γ 1 ( u γ/2 ) 2 dx Ω C(infu 0 )H α [u] 0 Proof: Systematic integration by parts (A.J.-Matthes 2006) Discrete case: If α = 2β then Hα[u d k ] nonincreasing If 1-D and uniform grid, Hα[u d k ] Hα[u d 0 ]e λt k

11 Implicit Euler finite-volume scheme 11 Numerical example t u = u β, no-flux b.c., H α [u] = u α dx Ω ( Ω ) α udx 0 5 α =.5 α = 1 5 α = 2 α = 6 10 α =.5 α = 1 α = 2 α = β = 1 2 : logh α[u(t)] versus t β = 2: logh α [u(t)] versus t 2-D scheme, uniform grid, initial data: Barenblatt profile Exponential time decay for all α

12 Overview 12 Introduction Implicit Euler finite-volume scheme Time-discrete one-leg multistep scheme Time-discrete Runge-Kutta scheme Extension

13 Semi-discrete multistep scheme 13 Equation: t u+a(u) = 0, t > 0, u(0) = u 0 Energy method: Let A satisfy A(u),u 0 1 d 2dt u 2 = t u,u = A(u),u 0 Entropy method: Let A(u),H (u) 0 1dH 2 dt [u] = tu,h (u) = A(u),H (u) 0 Entropy method generalizes from quadratic structure One-leg multistep scheme: τ 1 ρ(e)u k +A(σ(E)u k ) = 0, u k u(t k ) Approximation of t u(t k ): 1 τ ρ(e)uk = τ 1 p j=0 α ju k+j Approximation of u(t k ): σ(e)u k = p j=0 β ju k+j Question: H[u k ] generally not dissipated what can we do?

14 Semi-discrete multistep scheme 14 τ 1 ρ(e)u k +A(σ(E)u k ) = 0 ρ(e)u k = p j=0 α ju k+j, σ(e)u k = p j=0 β ju k+j Discrete energy method: Assume Hilbert space structure Conditions on (ρ, σ) yield second-order scheme Dahlquist 1963: (ρ,σ) A-stable p 2 Energy dissipation: If (ρ, σ) A-stable then G-stable, i.e., symmetric positive definite matrix (G ij ) such that (ρ(e)u k,σ(e)u k ) 1 2 Uk+1 2 G 1 2 Uk 2 G where U k = (u k,...,u k+p 1 ), U k 2 G = i,j G ij(u k+i,u k+j ) Implicit Euler: (u k+1 u k,u k+1 ) 1 2 uk uk 2 Energy dissipation: (Hill 1997) 1 2 Uk+1 2 G 1 2 Uk 2 G τ(a(σ(e)u k ),σ(e)u k ) 0

15 Semi-discrete multistep scheme 15 Discrete entropy method: Aim: Develop entropy-dissipative one-leg multistep scheme Difficulty: Energy dissipation based on quadratic 1 2 u 2 G Key idea: Enforce quadratic structure by v 2 = H(u) t u+a(u) = 0 H(u) 1/2 H (u) 1 t v A(u) = 0 Semi-discrete scheme: H(w k ) 1/2 H (w k ) 1 ρ(e)v k + τ 2 A(wk ) = 0 w k = H 1 ((σ(e)v k ) 2 ) Let H(u) = u α, α 1: ρ(e)v k +τb(σ(e)v k ) = 0, B(v) = α 2 v1 2/α A(v 2/α ) Is the scheme well-posed? Yes, under conditions on A Entropy dissipativity & positivity preservation? Yes! Numerical convergence order? Maximal order two

16 Semi-discrete multistep scheme 16 ρ(e)v k +τb(σ(e)v k ) = 0, B(v) = α 2 v1 2/α A(v 2/α ) Proposition (Entropy dissipation): Let (ρ, σ) be G-stable. Then H[V k ] = 1 2 V k 2 G with V k = (v k,...,v k+p 1 ) is nonincreasing in k. (Recall that (σ(e)v k ) 2/α u(t k ).) Proof: Use G-stability and assumption A(w k ),H (w k ) 0. Theorem (Convergence rate): Let (ρ, σ) be G-stable and of second order. Let u be smooth, B +κid be positive. Then, for τ > 0 small, v k u(t k ) α/2 Cτ 2. Proof: Use idea of Hundsdorfer/Steininger Well-posedness? Depends on specific operator A. Examples.

17 Semi-discrete multistep scheme 17 Population model (Shigesada-Kawasaki-Teramoto 1979) Motivation: Models segregation of population species Population densities: u 1, u 2, periodic boundary conditions t u 1 div((d 1 +a 1 u 1 +u 2 ) u 1 +u 1 u 2 ) = 0 t u 2 div((d 2 +a 2 u 2 +u 1 ) u 2 +u 2 u 1 ) = 0 Theorem: (A.J.-Milišić, NMPDE 2015) Let d 3, 1 < α < 2, 4a 1 a 2 max{a 1,a 2 }+1, (ρ,σ) G-stable. Then solution (v1,v k 2,w k 1,w k 2) k W 1,3/2 (T d ) such that wj k, σ(e)vk j 0 and H[V k+1 ]+ 2τ 2 α 2(α 1) d j (wj) k α/2 2 dx H[V k ] T d j=1 Scheme is nonnegativity-preserving and entropy-dissipative

18 Semi-discrete multistep scheme 18 Quantum diffusion model t u+ 2 : (u 2 logu) = 0 in T d, u(0) = u 0 Theorem: (A.J.-Milišić, NMPDE 2015) Let d 3, 1 < α < ( d+1) 2 d+2, (ρ,σ) G-stable. Then solution (v k,w k ) with (w k ) α/2 H 2 (T d ), w k,σ(e)v k 0 to 2 ατ (wk ) 1 α/2 1 ρ(e)v k + 2 : (w k 2 logw k ) = 0 in T d satisfying discrete entropy inequality H[V k+1 ]+ ατ 2 κ α ( (w k ) α/2 ) 2 dx H[V k ] T d Scheme is positivity-preserving and entropy-dissipative Conclusion: Method works well if u α, ulogu are entropies. Drawback: Discrete entropy not natural. Other time approx.?

19 Overview 19 Introduction Implicit Euler finite-volume scheme Time-discrete one-leg multistep scheme Time-discrete Runge-Kutta scheme Extension

20 Semi-discrete Runge-Kutta scheme 20 Equation: t u+a(u) = 0, t > 0, u(0) = u 0 Entropy: H[u] = Ω uα dx Discretization: s ( u k+1 = u k +τ b i K i, K i = A u i +τ i=1 Objective: Show that H[u k+1 ] H[u k ] τα Ω s j=1 a ij K j ) (u k+1 ) α 1 A(u k+1 )dx 0 Idea: Fix u := u k+1, interpret u k = v(τ) Define G(τ) = H[u] H[v(τ)] and take τ > 0 small: ( G(τ) = G(0) }{{} +τg (0)+ τ2 G 2 }{{ (0) } + τ ) 3} G {{ (ξ) } τg (0) =0 <0 < = τα u α 1 A(u)dx Ω To show: G (0) 0, G (0) < 0, G (ξ) <

21 Semi-discrete Runge-Kutta scheme 21 Example: Porous-medium eq. A = (u β ), show G (0) < 0 [ G (0)= α βu β 1 (u α 1 ) (u β )+(α 1)u α 1 ( (u β )) 2] dx T d Key idea: Systematic integration by parts (A.J.-Matthes 2006) Formulate G (0) in terms of ξ G = u u, ξ L = u u : G (0) = u α+2β 2 P(ξ G,ξ L )dx, P(ξ G,ξ L ) polynomial T d Interpret integrations by parts as polynomial manipulation: 0 = div(u α+2β 2 u 2 u)dx = u α+2β 2 T 1 (ξ G,ξ L )dx T d T d 0 = div(u α+2β 2 (D 2 u ui) u)dx = u α+2β 2 T 2 dx T d T d Find c 1, c 2 R such that for all ξ G, ξ L R: P(ξ G,ξ L )+c 1 T 1 (ξ G,ξ L )+c 2 T 2 (ξ G,ξ L ) > 0

22 Semi-discrete Runge-Kutta scheme 22 t u = (u β ) in T d, t > 0, u(0) = u 0 Theorem: (A.J.-Schuchnigg, work in progress) There exists a parameter range for (α,β) such that any implicit Runge-Kutta schema dissipates the entropy: H[u k+1 ] = (u k+1 ) α dx H[u k ], τ > 0 small T d F[u k+1 ] = 1 (u α/2 ) x 2 dx F[u k ], only 1-D 2 T

23 Overview 23 Introduction Implicit Euler finite-volume scheme Time-discrete one-leg multistep scheme Time-discrete Runge-Kutta scheme Extension

24 Extension 24 Higher-order minimizing movement scheme: Restrict to Hilbert space setting: t u = φ(u) First-order minimizing movement scheme: ( u k u k 1 = τ φ(u k ), u k = argmin 1 2τ uk 1 v 2 +φ(v) ) v H Gradient flow in the L 2 -Wasserstein distance (Ambrosio, Otto, Savaré,...) Higher-order minimizing movement scheme: one-leg meth. ( ρ(e)u k = τ φ(σ(e)u k ), w = argmin 1 2τ η +v 2 +φ(v) ) v H p 1 ( ) ( αj β η = p p 1 ) α p β j u k+j, u k+p = β 1 k w β j u k+j j=0 Result: Discrete entropy in G-norm dissipated (A.J.-Fuchs 14) Extensions: differ. inclusions, metric/wasserstein spaces? j=0

25 Summary 25 Discrete entropy dissipation for... (v k = (u k ) α/2 ) Numerical scheme Discrete entropy Implicit Euler, finite volume H[u k ] = K K (uk K )α + fully discrete, only first order One-leg multi-step scheme H[V k ] = Ω G ij(v k+i,v k+j ) + full theory (G-stability), entropy def. not natural, max. 2nd order Runge-Kutta scheme H[u k ] = Ω (uk ) α dx + natural entropy definition, any order, only semi-discrete High-order minimizing movem. H[V k ] = Ω G ij(v k+i,v k+j ) + natural for gradient flows, entropy not natural, max. second order Open problem: find discrete calculus for systematic discrete summation by parts