Exponential integrators for semilinear parabolic problems

Transcription

1 Exponential integrators for semilinear parabolic problems Marlis Hochbruck Heinrich-Heine University Düsseldorf Germany Innsbruck, October 2004 p.

2 Outline Exponential integrators general class of methods order conditions for stiff problems explicit exponential Runge-Kutta methods of order 4 summary Implementation preconditioned Krylov subspace methods for matrix functions summary and outlook p.2

3 Exponential integrators recent interest in exponential integrators for semilinear problems u + Au = g(t, u) Cox and Matthews, JCP 2002 Trefethen and Kassam, to appear in SISC Krogstad, Thesis and Preprint 2003 Minchev, Thesis 2004 Berland, Owren, and Skaflestad, Preprint 2004 Hochbruck and Ostermann, APNUM 2004, Preprint 2004 p.3

4 Exponential integrators, cont d numerical experiments in these references: show that exponential integrators outperform standard methods for a number of examples implementation only for small problems or problems with periodic boundary conditions in this talk: present error bounds discuss implementation p.4

5 PART I Exponential Integrators joint work with Alexander Ostermann University of Innsbruck, Austria p.5

6 General methods u + Au = g(u) motivated by the construction of collocation methods u n+ = χ( ha)u n + h U n,i = χ i ( ha)u n + h s b i ( ha)g n,i, G n,i = g(u n,i ) i= s a ij ( ha)g n,j j= (Friedli 978; Strehmel, Weiner 987) A 0: Runge-Kutta method with b i = b i (0), a ij = a ij (0) consistency: χ(0) = χ i (0) = explicit methods: χ (z), a ij (z) 0, i j s p.6

7 General properties simplifying assumptions (preservation of equilibria) s j= a ij (z) = χ i(z) z, s i= b i (z) = χ(z) z equivalent numerical scheme s u n+ = u n + h U n,i = u n + h G n,i = g(u n,i ) i= s j= b i ( ha) ( G n,i Au n ), a ij ( ha) ( G n,j Au n ), p.7

8 for small A 0 Friedli 978 Strehmel, Weiner 987 Order conditions bi-colored trees, same trees as for W-methods (Berland, Owren, Skaflestad, 2004) up to desired order χ(z) = e z, χ i (z) = e c iz b i, a ij linear combination of ϕ,..., ϕ s ϕ j ( ta) = (j )!t j t 0 e (t τ)a τ j dτ, j p.8

9 for small A 0 Friedli 978 Strehmel, Weiner 987 Order conditions bi-colored trees, same trees as for W-methods (Berland, Owren, Skaflestad, 2004) up to desired order χ(z) = e z, χ i (z) = e c iz b i, a ij linear combination of ϕ,..., ϕ s ϕ j ( ta) = (j )!t j t 0 e (t τ)a τ j dτ, j these order conditions are not sufficient for error bounds p.8

10 Analytical framework u (t) + Au(t) = g(u) (X, ) Banach space, D(A) domain of A in X Assumption on A (Henry, Pazy): A : D(A) X X sectorial: A densely defined, closed linear operator on X, satisfying resolvent condition λ (λi A) M λ a a ϑ for à = A + ωi, ω > a, fractional powers Ãα well defined p.9

11 Analytical framework, II Assumption on g: V = D(Ãα ), v V = Ãα v X g(v) g(w) X L v w V framework = stability φ( ta) X X + φ( ta) V V + t γ Ã γ φ( ta) X X C, for 0 γ, φ = exp, ϕ i, b i, a ij in this talk: X = C(Ω), =, D(A) = C 2 (Ω) C 0 (Ω) here α = 0 p.0

12 Error analysis insert exact solution into numerical scheme u(t n + c i h) = e ciha u(t n ) + h a ij ( ha)f(t n + c j h) + n,i, u(t n+ ) = e ha u(t n ) + h b i ( ha)f(t n + c i h) + δ n+ defects n,i = j h j ψ j,i ( ha)f (j ) (t n ) +... ψ j,i ( ha) = ϕ j ( c i ha) c }{{} =: ϕ j,i j i s k= a ik ( ha) c j k (j )! and analogous for δ n+ and ψ j p.

13 Stiff order conditions No. order order condition ψ ( ha) = ψ 2 ( ha) = ψ i ( ha) = ψ 3 ( ha) = i b i( ha) J ψ 2,i ( ha) = ψ 4 ( ha) = i b i( ha) J ψ 3,i ( ha) = i b i( ha) J j a ij( ha) J ψ 2,j ( ha) = i b i( ha)c i K ψ 2,i ( ha) = 0 p.2

14 Main result Theorem (H, Ostermann 2004) Assume stiff order conditions are satisfied up to order p, 2 p 4 and that ψ p (0) = 0. Further assume that the remaining conditions of order p are satisfied in the weaker form with Then b i (0) instead of b i ( ha), 2 i s. u n u(t n ) Ch p, where C is independent of n and h. p.3

15 Main result Theorem (H, Ostermann 2004) Assume stiff order conditions are satisfied up to order p, 2 p 4 and that ψ p (0) = 0. Further assume that the remaining conditions of order p are satisfied in the weaker form with Then b i (0) instead of b i ( ha), 2 i s. u n u(t n ) Ch p, where C is independent of n and h. Replacing A by 0 elsewhere in the order conditions leads to order reductions in general p.3

16 Second order methods order conditions for order two: b ( ha) + b 2 ( ha) = ϕ ( ha) order b 2 ( ha)c 2 = ϕ 2 ( ha) order 2 a 2 ( ha) = c 2 ϕ ( c 2 ha) order 2 yields one-parameter family of second order methods: 0 c 2 c 2 ϕ ( c 2 ha) ϕ ( ha) c 2 ϕ 2 ( ha) c2 ϕ 2 ( ha) 2nd condition can be weakened to b 2 (0)c 2 = ϕ 2 (0) = 2 p.4

17 Second order methods, cont d one-parameter family using only ϕ : 0 c 2 c 2 ϕ ( c 2 ha) ( 2c 2 )ϕ ( ha) 2c2 ϕ ( ha) most attractive choice: c 2 = 2 which yields b = 0 exponential Runge methods p.5

18 Higher order methods order three conditions (no. 5) for s = 3: b 2 ( ha) J c 2 2ϕ 2 ( c 2 ha)+b 3 ( ha) J ψ 2,3 ( ha) = 0 for all J where ψ 2,3 ( ha) = c 2 3ϕ 2 ( c 3 ha) c 2 a 32 ( ha) p.6

19 Higher order methods order three conditions (no. 5) for s = 3: b 2 ( ha) J c 2 2ϕ 2 ( c 2 ha)+b 3 ( ha) J ψ 2,3 ( ha) = 0 for all J where ψ 2,3 ( ha) = c 2 3ϕ 2 ( c 3 ha) c 2 a 32 ( ha) possible solutions b 2 = 0, then ψ 2,3 ( ha) = 0 b 2 = γb 3, then γc 2 2ϕ 2 ( c 2 ha) + ψ 2,3 ( ha) = 0 p.6

20 4-stage explicit exponential methods consider only methods which reduce to classical Runge-Kutta method for A = 0: several options for designing exponential methods we study methods known to perform well in experiments method of Cox and Matthews (2002) method of Krogstad (2003) p.7

21 Order conditions observations taken from order conditions any fourth order method has to involve ϕ, ϕ 2, and ϕ 3 n,2 = O(h 2 ) for any explicit method since b i (0) 0 for all i we have to duplicate nodes to get cancellation of defects in U n,2, U n,3, U n,4 p.8

22 Cox and Matthews method (ETDRK4) ϕ, ϕ,3 2 ϕ,3 ( ϕ0,3 ) 0 ϕ,3 ϕ 3ϕ 2 + 4ϕ 3 2ϕ 2 4ϕ 3 2ϕ 2 4ϕ 3 4ϕ 3 ϕ 2 satisfies (from total of 9 conditions, 6 9 are for order 4) conditions 4, and 6 (ψ 4 (0) = 0) only weakened form of conditions 5 and 9 (arguments of b i evaluated for A = 0) very weak form of conditions 7 and 8 (all arguments evaluated for A = 0) p.9

23 Cox and Matthews method (ETDRK4) Theorem for periodic boundary conditions: order 4 in the worst case: order 2 if A γ J, J = g is bounded: order 2 + γ u p.20

24 Krogstad s method (ETDRK4-B) ϕ,2 2 ϕ,3 ϕ 2,3 ϕ 2,3 ϕ,4 2ϕ 2,4 0 2ϕ 2,4 ϕ 3ϕ 2 + 4ϕ 3 2ϕ 2 4ϕ 3 2ϕ 2 4ϕ 3 ϕ 2 + 4ϕ 3 satisfies (from total of 9 conditions, 6 9 are for order 4) conditions 5, 9, and 6 (ψ 4 (0) = 0) very weak form of conditions 7 and 8 (all arguments evaluated for A = 0) p.2

25 Krogstad s method (ETDRK4-B) Theorem for periodic boundary conditions: order 4 in the worst case: order 3 if A γ J, J = g is bounded: order 3 + γ u p.22

26 Krogstad s method (ETDRK4-B) Theorem for periodic boundary conditions: order 4 in the worst case: order 3 if A γ J, J = g is bounded: order 3 + γ u is it possible to obtain full order 4? p.22

27 Full order four Runge-Kutta-type methods no 4-stage method of order 4 exists p.23

28 Full order four Runge-Kutta-type methods no 4-stage method of order 4 exists new 4th order method with s = 5, c 5 = / ϕ,2 2 ϕ,3 ϕ 2,3 ϕ 2,3 ϕ,4 2ϕ 2,4 ϕ 2,4 ϕ 2,4 2 2 ϕ,5 2a 5,2 a 5,4 a 5,2 a 5,2 4 ϕ 2,5 a 5,2 ϕ 3ϕ 2 + 4ϕ ϕ 2 + 4ϕ 3 4ϕ 2 8ϕ 3 with a 5,2 = 2 ϕ 2,5 ϕ 3,4 + 4 ϕ 2,4 2 ϕ 3,5 p.23

29 Numerical experiment I test problem: u u xx = +u + g(t), Dirichlet b.c. 2 with exact solution x( x)e t, x = /00, L -norm errors 0 0 Runge Heun 0 2 Krogstad Cox Matthews in this case, γ = p.24

30 Numerical experiment I test problem: u u xx = +u + g(t), Dirichlet b.c. 2 with exact solution x( x)e t, x = /200, L -norm errors 0 0 Runge Heun 0 2 Krogstad Cox Matthews in this case, γ = p.24

31 Numerical experiment II test problem: u u xx = 0 u(x, t)dx, Dirichlet b.c. x = /200, L -norm errors 0 0 new method Krogstad 0 2 CoxMatthews h 2.5 h h in this case, γ = / p.25

32 Summary presented error bounds for explicit exponential Runge-Kutta methods for abstract odes u + Au = g(t, u) new order conditions for stiff problems verified sharpness of these bounds numerically References: M. Hochbruck, A. Ostermann, Exponential Runge-Kutta methods for parabolic problems, to appear in APNUM 2004 M. Hochbruck, A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, Preprint 2004 p.26

33 PART II Implementation joint work with Jasper van den Eshof University of Düsseldorf, Germany p.27

34 Approximation of matrix operators assume A symmetric, positive semidefinite (but A sectorial works as well) yesterday: showed that Krylov approximations of exp( ha)b and ϕ j ( ha)b in each step: one matrix-vector multiplication Ax always converge superlinearly superlinear convergence starts at m ha /2 p.28

35 Approximation of matrix operators assume A symmetric, positive semidefinite (but A sectorial works as well) yesterday: showed that Krylov approximations of exp( ha)b and ϕ j ( ha)b in each step: one matrix-vector multiplication Ax always converge superlinearly superlinear convergence starts at m ha /2 want to apply preconditioning in order to get mesh independent convergence (as for multigrid) p.28

36 A is discretization of elliptic PDE eigenvalues of D Laplacian not of importance for result important eigenvalues exponential function is rapidly increasing only smallest eigenvalues of A are important but these are hard to find by Lanczos (Kuijlaars 2000) p.29

37 A is discretization of elliptic PDE eigenvalues of D Laplacian not of importance for result important eigenvalues exponential function is rapidly increasing only smallest eigenvalues of A are important but these are hard to find by Lanczos (Kuijlaars 2000) Idea: spectral transformation (I + γa) p.29

38 Preconditioning Lanczos approximations Idea: Lanczos process w.r.t. (I + γa) (instead of A) in each step: one linear system solve (I + γa)x = b obtain basis V m, tridiagonal T m approximation y m (τ) = V m exp( τ T m )e, Tm = γ (T m I). Related work by Moret and Novati, 2002 p.30

39 Questions optimal choice of shift parameter γ a posteriori error estimates iterative solution of linear systems as inner iteration of Lanczos process (preconditioned conjugate gradient iteration) Convergence: A priori: mesh independent but sublinear convergence Table to determine optimal γ for desired accuracy (computed with Remez algorithm) p.3

40 Example, D periodic Poisson, dimension 0 5 y m (τ) = V m exp( τ T m )e, γ = τ/0 0 0 Error Iteration Error for τ = /2, τ = /20, τ = /50, τ = /2000 and upper bound (dotted). p.32

41 Numerical experiment: mesh independence d 2 dx 2 + d2 dy 2 discretized on uniform grids, ɛ = 0 8, γ = τ/0 shifted systems solved by SAMG Fortran package dimension Fixed τ = /0 65/3 70/3 78/3 78/3 90/3 9/3 τ = /00 04/5 48/6 89/8 90/8 08/8 08/8 Relaxed τ = /0 43/3 48/3 52/3 53/3 58/3 60/3 τ = /00 62/5 32/6 48/8 55/8 6/8 64/8 Table reports V-cycles/Lanczos iterations p.33

42 Summary Generalized concept for preconditioning matrix functions spectral transformation Worst case convergence bounded by restricted rational approximations Optimal linear solver optimal method Relaxation of inner iteration results in gain up to 40 percent Oscillatory problems? Reference: J. van den Eshof, M. Hochbruck Preconditioning Lanczos approximations to the matrix exponential, submitted to SISC, March 2004 p.34