Discrete entropy methods for nonlinear diffusive evolution equations - PowerPoint PPT Presentation

1 Discrete entropy methods for nonlinear diffusive evolution equations Ansgar J¨ ungel Vienna University of Technology, Austria www.jungel.at.vu Joint work with E. Emmrich (TU Berlin), M. Bukal (Zagreb), J.-P. Miliˇ si´ c (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 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 2 [ u ] ≥ − κdH dt [ u ] ⇒ D [ u ] = − dH dt [ u ] ≥ κH [ u ] Consequences: dt ≤ − κH implies that H [ u ( t )] ≤ H [ u (0)] e − κt ∀ t > 0 • dH • H [ u ] ≤ κ − 1 D [ u ] corresponds to convex Sobolev inequality

3 Introduction Example: heat equation ∂ t u = ∆ u on torus T d � Entropy: H [ u ] = Ω u log( u/u ∞ ) dx , u ∞ : steady state � 1 Entropy-dissipation inequality: � T d |∇√ u | 2 dx ≥ 0 D [ u ] = − dH dt [ u ] = 4 � 2 Second-order time derivative: � d 2 H ∆ √ u √ u ∆ udx ≥ − κdH ⇒ dH dt 2 [ u ] = 4 dt ≤ − κH dt T d • Exponential decay to equilibrium: � u ∞ dx ≤ H [ u (0)] e − κt u H [ u ( t )] = u log Ω • Logarithmic Sobolev inequality: � � T d |∇√ u | 2 dx u ∞ dx ≤ 1 κD [ u ] = 4 u H [ u ] = u log κ Ω Benefit: very robust, in particular for nonlinear problems

4 Introduction Setting: ∂ t u + A ( u ) = 0 , t > 0 , u (0) = u 0 Task: Develop discrete entropy methods Program: τ ( u k − u k − 1 ) + A ( u k ) = 0 • Implicit Euler scheme: 1 t u k + A ( u k , u k − 1 , . . . ) = 0 • Higher-order time scheme: ∂ τ • 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 Setting: ∂ t u + A ( u ) = 0 , t > 0 , u (0) = u 0 Task: Develop discrete entropy methodsg Program: τ ( u k − u k − 1 ) + A ( u k ) = 0 ✔ Implicit Euler scheme: 1 t u k + A ( u k , u k − 1 , . . . ) = 0 ✔ Higher-order time scheme: ∂ τ ✔ 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 • Introduction • Implicit Euler finite-volume scheme • Semi-discrete one-leg multistep scheme • Semi-discrete Runge-Kutta scheme

7 Implicit Euler finite-volume scheme Example: ∂ t u = ∆ u β with no-flux boundary conditions � � Ω u α dx − ( Ω udx ) α Continuous case: entropy H α [ u ] = � � dH α dt = d u α dx = α u α − 1 ∆ u β dx dt Ω Ω � 4 αβ |∇ 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 � ≤ C B �∇ f � q | f | q dx − | f | 1 /p dx L 2 (Ω) , q ≤ 2 Ω Ω Standard Beckner inequality: q = 2 Proof: Differentiate L p interpolation inequality (Dolbeault) and use generalized Poincar´ e-Wirtinger inequality Task: Translate computations to discrete case

8 Implicit Euler finite-volume scheme Finite-volume scheme: Ω = ∪ K σ • Control volumes K , edges σ = K | L x L x K d σ K • Transmissibility coeff.: τ σ = | K | /d σ L � K ) β − ( u k | K | ( u k K − u k − 1 τ σ (( u k L ) β ) = 0 K ) + τ σ = K | L α [ u ] = � K − ( � Discrete case: H d K | K | u α K | K | u K ) α � K ) α − ( u k − 1 H d α [ u k ] − H d α [ u k − 1 ] = | K | (( u k K ) α ) K � | K | ( u k K ) α − 1 ( u k K − u k − 1 ≤ K ) K ≤ − C 1 | ( u k ) ( α + β − 1) / 2 | 2 H 1 ≤ − C 2 H d α [ u k ] ( α + β − 1) /α Follows from discrete Beckner inequality Proof: Use discrete Poincar´ e-Wirtinger inequality (Bessemoulin-Chatard, Chainais-Hillairet, Filbet 2012)

9 Implicit Euler finite-volume scheme 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 � Ω |∇ u α/ 2 | 2 dx First-order entropies? H α [ u ] = • Continuous case: Let ( α, β ) ∈ M d 4 � dH α div ( u α/ 2 − 1 ∇ u α/ 2 )∆ u β dx 3 dt = − α Ω � 2 u α + β − γ − 1 (∆ u γ/ 2 ) 2 dx ≤ − C 1 Ω ≤ − C (inf u 0 ) H α [ u ] 0 0 2 4 6 8 Proof: Systematic integration by parts (A.J.-Matthes 2006) • Discrete case: If α = 2 β then H d α [ u k ] nonincreasing If 1-D and uniform grid, H d α [ u k ] ≤ H d α [ u 0 ] e − λt k

10 Implicit Euler finite-volume scheme Numerical results: Zeroth-order entropies �� � α � ∂ t u = ∆ u β , u α dx − H α [ u ] = udx Ω Ω α = . 5 α = . 5 0 α = 1 − 5 α = 1 α = 2 α = 2 α = 6 α = 6 − 10 − 5 − 15 − 10 − 20 − 15 − 25 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 0 . 2 0 . 4 0 . 6 0 . 8 β = 1 2 : log H α [ u ( t )] versus t β = 2 : log H 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 Numerical results: First-order entropies � ∂ t u = ∆ u β , |∇ u α/ 2 | 2 dx H α [ u ] = Ω 5 α = . 5 α = . 5 5 α = 1 α = 1 0 α = 2 α = 2 α = 6 α = 6 0 − 5 − 5 − 10 − 15 − 10 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 0 . 2 0 . 4 0 . 6 0 . 8 β = 1 2 : log H α [ u ( t )] versus t β = 2 : log H α [ u ( t )] versus t • 2-D scheme, uniform grid, initial data: truncated polynom. • Exponential time decay for all α

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

13 Semi-discrete multistep scheme Equation: ∂ t u + A ( u ) = 0 , t > 0 , u (0) = u 0 “Energy” method: Let A satisfy � A ( u ) , u � ≥ 0 1 d dt � u � 2 = � ∂ t u, u � = −� A ( u ) , u � ≤ 0 2 “Entropy” method: Let � A ( u ) , H ′ ( u ) � ≥ 0 1 dH dt [ u ] = � ∂ t u, H ′ ( u ) � = −� A ( u ) , H ′ ( u ) � ≤ 0 2 → 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 ) τ ρ ( E ) u k = 1 � p • Approximation of ∂ t u ( t k ) : 1 j =0 α j u k + j τ • Approximation of u ( t k ) : σ ( E ) u k = � p j =0 β j u k + j Question: H [ u k ] generally not dissipated – what can we do?

14 Semi-discrete multistep scheme Discrete “energy” method: Assume Hilbert space structure τ − 1 ρ ( E ) u k + A ( σ ( E ) u k ) = 0 p p � � ρ ( E ) u k = σ ( E ) u k = α j u k + j , β j u k + j j =0 j =0 • 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 � U k +1 � 2 G − 1 2 � U k � 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 � U k +1 � 2 G − 1 2 � U k � 2 G ≤ − τ ( A ( σ ( E ) u k ) , σ ( E ) u k ) ≤ 0

15 Semi-discrete multistep scheme 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 + 1 2 A ( u ) = 0 Semi-discrete scheme: H ( w k ) 1 / 2 H ′ ( w k ) − 1 ρ ( E ) v k + τ 2 A ( w k ) = 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 v 1 − 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 ρ ( E ) v k + τB ( σ ( E ) v k ) = 0 , B ( v ) = α 2 v 1 − 2 /α A ( v 2 /α ) Proposition (Entropy dissipation): Let ( ρ, σ ) be G-stable. G with V k = ( v k , . . . , v k + p − 1 ) is Then H [ V k ] = 1 2 � V k � 2 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 ) α/ 2 2 � A ( w k ) , H ′ ( w k ) � ≤ 0 = τ Theorem (Convergence rate): Let ( ρ, σ ) be G-stable and of second order. Let u be smooth, B + κ Id 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