Discrete entropy methods for nonlinear diffusive evolution equations

Transcription

1 Discrete entropy methods for nonlinear diffusive evolution equations 1 Ansgar Jüngel Vienna University of Technology, Austria Joint work with E. Emmrich (TU Berlin), M. Bukal (Zagreb), J.-P. Milišić (Zagreb), C. Chainais-Hillairet (Lille), S. Schuchnigg (Vienna) Continuous and discrete entropy methods Implicit Euler finite-volume scheme Higher-order time schemes Extensions

2 Introduction 2 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

3 Introduction 3 Example: heat equation t u = u on torus T d Entropy: H[u] = Ω ulog(u/u )dx, u : steady state 1 Entropy-dissipation inequality: D[u] = dh dt [u] = 4 T d u 2 dx 0 2 Second-order time derivative: d 2 H dt [u] = 4 2 u u udx κ dh dh T d dt dt κh Exponential decay to equilibrium: H[u(t)] = ulog u u dx H[u(0)]e κt Ω Logarithmic Sobolev inequality: H[u] = ulog u u dx 1 Ω κ D[u] = 4 u 2 dx κ T d Benefit: very robust, in particular for nonlinear problems

4 Introduction 4 Setting: t u+a(u) = 0, t > 0, u(0) = u 0 Task: Develop discrete entropy methods Program: Implicit Euler scheme: 1 τ (uk u k 1 )+A(u k ) = 0 Higher-order time scheme: τ tu k +A(u k,u k 1,...) = 0 Finite-volume scheme: t u K +A(u K ) = 0, u K : const., K: control volume Fully discrete schemes, higher-order spatial discretizations Higher-order minimizing movement schemes Questions: Is H[u k ] dissipated? Rate of entropy decay? Key idea: Translate entropy method to discrete settings

5 Introduction 5 Setting: t u+a(u) = 0, t > 0, u(0) = u 0 Task: Develop discrete entropy methodsg Program: Implicit Euler scheme: 1 τ (uk u k 1 )+A(u k ) = 0 Higher-order time scheme: τ tu k +A(u k,u k 1,...) = 0 Finite-volume scheme: t u K +A(u K ) = 0, u K : const., K: control volume Fully discrete schemes, higher-order spatial discretizations Higher-order minimizing movement schemes (in progress) Questions: Is H[u k ] dissipated? Rate of entropy decay? Key idea: Translate entropy method to discrete settings

6 Overview 6 Introduction Implicit Euler finite-volume scheme Semi-discrete one-leg multistep scheme Semi-discrete Runge-Kutta scheme

7 Implicit Euler finite-volume scheme 7 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 Standard Beckner inequality: q = 2 Ω Proof: DifferentiateL p interpolationinequality(dolbeault) and use generalized Poincaré-Wirtinger inequality Task: Translate computations to discrete case

8 Implicit Euler finite-volume scheme 8 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

9 Implicit Euler finite-volume scheme 9 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

10 Implicit Euler finite-volume scheme 10 Numerical results: Zeroth-order ( entropies t u = u β, H α [u] = u α dx udx Ω Ω ) α 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

11 Implicit Euler finite-volume scheme 11 Numerical results: First-order entropies t u = u β, H α [u] = u α/2 2 dx Ω 5 α =.5 α = 1 α = α =.5 α = 1 α = 2 0 α = 6 5 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: truncated polynom. Exponential time decay for all α

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

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 = 1 τ 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 Discrete energy method: Assume Hilbert space structure τ 1 ρ(e)u k +A(σ(E)u k ) = 0 p ρ(e)u k = α j u k+j, σ(e)u k = j=0 p j=0 β j u k+j 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 ) 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: By G-stability and assumption on A, H[V k+1 ] H[V k ] = 1 2 V k+1 2 G 1 2 V k 2 G (ρ(e)v k,σ(e)v }{{} k ) =(w k ) = τ α/2 2 A(wk ),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, and p = 2. Then, for τ > 0 small, v k u(t k ) α/2 Cτ 2. Proof: Use idea of Hundsdorfer/Steininger 1991

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 2014, to appear) 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 2014, to appear) 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,ρ(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 and ulogu are entropies and entropy dissipation gives Sobolev estimates

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

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 G (0) = τα u α 1 A(u)dx, A diff. operator order p Ω G (0) = α u α 2[ uda(u)(a(u)) (α 1)(A(u)) 2] dx Ω G (0) involves derivatives of order p G (0) involves derivatives of order 2p Key idea: Systematic integration by parts (A.J.-Matthes 2006) Example: t u = (u β ) in T d G (0) = α u α 1 (u β )dx T d = α(α 1)β u α+β 3 u 2 dx 0 T [ d G (0) = α βu β 1 (u α 1 ) (u β )+(α 1)u α 1 ( (u β )) 2] dx T d 21

22 Semi-discrete Runge-Kutta scheme [ G (0) = α βu β 1 (u α 1 ) (u β )+(α 1)u α 1 ( (u β )) 2] dx T d Systematic integration by parts: 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 ) = P(ξ G,ξ L )+c 1 T 1 (ξ G,ξ L )+c 2 T 2 (ξ G,ξ L ) > 0 22

23 Semi-discrete Runge-Kutta scheme 23 Polynomial decision problem: Solve by quantifier elimination c 1,c 2 : ξ G,ξ L : P(ξ G,ξ L ) = (P +c 1 T 1 +c 2 T 2 )(ξ G,ξ L ) > 0 Tarski 1930: Such quantified statements can be reduced to a quantifier-free statement in an algorithmic way. + Implementations in Mathematica, QEPCAD available + Gives complete, exact answer and proof Algorithms are doubly exponential in no. of ξ i, c i Consequence: G (0) = T d u 2α+β 2 Pdx < 0 G(τ) = G(0)+τG(0)+ 1 2 τ2 G (0)+ 1 6 τ3 G (ξ) τg (0) G(τ) = H[u k+1 ] H[u k ] τg (0) = α(α 1)βτ (u k+1 ) α+β 3 u k+1 2 dx 0 T d

24 Semi-discrete Runge-Kutta scheme 24 t u = (u β ) in T d, t > 0, u(0) = u 0 Theorem: (A.J.-Schuchnigg, 2014) 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

25 Intermediate summary 25 Implicit Euler finite-volume scheme: Discrete exponential/algebraic equilibration rates Tools: Compute discrete dh dt, use discrete Beckner ineq. Higher-order time schemes: One-leg multistep: G-norm is dissipative, up to order two Tools: Enforce quadratic structure, G-stability theory Implicit Runge-Kutta: Discrete entropy is dissipative Tools: Taylor expansion, systematic integration by parts Questions: What about spatially discrete Bakry-Emery approach? What about higher-order minimizing movement scheme?

26 Extensions 26 Question: What about discrete Bakry-Emery? Mielke 2013: Geodesic λ-convexity for discrete Fokker-Planck ( ) u t u = div( u+u φ) = div u u log u u, u = ce φ Discrete entropy: H[u] = i u ilog u i u,i Finite-volume discretization: uniform 1-D mesh with size x t u = S LSlog u u, L = diag(l i ) S, S: discrete divergence, gradient respectively L i = u,i u,i+1 Λ( u i u,i, u i+1 u,i+1 ) and Λ(a,b) = a b loga logb 1 Theorem: If (φ ( x) 2 i+1 2φ i +φ i 1 ) 2κ > 0 then d 2 H dt 4 ) dh exp. decay 2 ( x) 2(1 e κ( x)2 dt Asymptotically sharp rate. Extension to nonlinear eqs.?

27 Extensions 27 Question: Higher-order minimizing movement scheme? Restrict first 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 scheme ( 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 Discrete entropy in G-norm is dissipated (A.J.-Fuchs 2014) Extensions: differ. inclusions, metric/wasserstein spaces? j=0

28 Summary 28 Implicit Euler finite-volume scheme: Exponential/algebraic decay of discrete entropy H[u k ] (Tool: Discrete generalized Beckner inequality) Higher-order one-leg multistep scheme: Discrete entropy in G-norm dissipated (Tool: G-stability theory of Dahlquist) Higher-order Runge-Kutta scheme: Discrete entropy H[u k ] dissipated (Tool: Systematic integration by parts) Discrete Bakry-Emery method: Exponential entropy decay & discrete convex Sobolev ineq. Higher-order minimizing movement schemes: Discrete entropy in G-norm is dissipated