Exponential multistep methods of Adams-type Marlis Hochbruck and Alexander Ostermann KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT) 0 KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu
Outline Motivation Exponential Adams methods Linearized exponential multistep methods Implementation Application: Simulation of optical resonators 1
Motivation general initial value problem u (t) = F ( t, u(t) ), u(t 0 ) = u 0. (in this talk: autonomous problems only) convergence analysis for semilinear initial value problems variation of constants formula u (t) = Au(t) + g ( t, u(t) ), u(0) = u 0 h u(t n+1 ) = e ha u(t n ) + e (h τ)a g ( u(t n + τ) ) dτ 0 t n = t 0 + nh, n = 0, 1,... idea of multistep methods: replace g by interpolation polynomial 2
Exponential Adams methods Given approximations u j u(t j ), consider interpolation polynomial p n through ( tn k+1, g(u n k+1 ) ),..., ( t n, g(u n ) ), given by k 1 p n (t n + θh) = G n + j=1 ( 1) j ( θ j where j G n denotes jth backward difference ) j G n, G j = g(u j ) 0 G n = G n, j G n = j 1 G n j 1 G n 1, j = 1, 2,.... replace nonlinearity g by p n in v.o.c formula 3
Exponential Adams methods, cont d numerical scheme h u n+1 = e ha u n + e (h τ)a p n (t n + τ)dτ 0 k 1 = u n + hϕ 1 ( ha)f(u n ) + h γ j ( ha) j G n j=1 with weights γ j (z) = ( 1) j 1 0 ( ) θ e (1 θ)z dθ, j 0. j Certaine, 1960, Nørsett 1969, Cox and Matthews 2002 rational variants: Lambert and Sigurdsson 1972, Verwer 1976 4
Exponential Adams methods, cont d in terms of ϕ-functions the weights γ j are ϕ j (z) = γ 1 = ϕ 2 1 e (1 θ)z 0 γ 2 = ϕ 3 + 1 2 ϕ 2 θ j 1 γ 3 = ϕ 4 + ϕ 3 + 1 3 ϕ 2 dθ, j 1, (j 1)! γ 4 = ϕ 5 + 3 2 ϕ 4 + 11 12 ϕ 3 + 1 4 ϕ 2 γ 5 = ϕ 6 + 2ϕ 5 + 7 4 ϕ 4 + 5 6 ϕ 3 + 1 5 ϕ 2 5
Examples of exponential Adams methods k = 1: exponential Euler method k = 2: u n+1 = u n + hϕ 1 ( ha)f(u n ) u n+1 = u n + hϕ 1 ( ha)f(u n ) + hϕ 2 ( ha) ( g(u n ) g(u n 1 ) ) interpretation as corrected exponential Euler step implementation can take advantage of j G n =O(h j ) H., Lubich, Selhofer 1998: exp4 methods are not invariant under linearization 6
Starting values use interpolation polynomial p in ( t0, g(u 0 ) ),..., ( t k 1, g(u k 1 ) ) within v.o.c. formula over interval of length mh for m = 1,..., k 1 approximate k 1 u m = u 0 + mhϕ 1 ( mha)f(u 0 ) + h σ m,l ( ha) l G 0, Calvo and Palencia, 2006 solve nonlinear system for u 1,..., u k 1 using fixed point iteration l=1 7
Linearized exponential multistep methods construction involves two steps linearize u = F(u) in each step at u n u(t n ) to get u (t) = J n u(t) + g n ( u(t) ) with J n = F u (u n), g n (u) = F(u) J n u apply explicit exponential integrator to linearized problem generalization to non-autonomous problems (not in this talk) H., Ostermann, Schweitzer (2006, 2009), Tokman (2006) 8
Examples k = 1: linearized exponential Euler method or exponential Rosenbrock Euler method u n+1 = u n + hϕ 1 (hj n )F(u n ) second-order convergent (H., Ostermann, Schweitzer, 2009) k = 2: Tokman 2006 u n+1 = u n + h ϕ 1 (hj n )F(u n ) 2h 3 ϕ 2(hJ n ) ( g n (u n ) g n (u n 1 ) ) third-order convergent (H., Ostermann, 2010) k = 2: u n+1 = u n + hϕ 1 (hj n )F(u n ) 2hϕ 3 (hj n ) ( g n (u n ) g n (u n 1 ) ) third-order convergent (H., Ostermann, 2010) 9
Linearized exponential multistep methods general construction of higher order methods v.o.c. formula applied to linearized ode yields exploit u (t) = J n u(t) + g n ( u(t)) h u(t n+1 ) = e hj n u(t n ) + e (h τ)j ( n g n u(tn + τ) ) dτ 0 g n u (u n) = 0, by approximating g n by Hermite interpolation polynomial p n of degree k satisfying p n (t n ) = 0 10
Linearized exponential Adams method, cont d Hermite interpolation polynomial p n of degree k interpolating in ( tn k+1, g n (u n k+1 ) ),..., ( t n, g n (u n ) ) and satisfying p n(t n ) = 0 k 1 p n (t n + θh) = G n,n + j=1 ( θ ( 1) j+1 θ j ) j 1 l l G n,n, l=1 where G n,m = g n (u m ) and j G n,m denotes the jth backward difference (w.r.t. m) 11
Linearized exponential Adams method, cont d inserting Hermite interpolation polynomial into v.o.c. formula with weights 1 u n+1 = e hj n u n + h e h(1 θ)j n p n (t n + θh)dθ 0 k 1 = u n + hϕ 1 (hj n )F(u n ) + h j=1 γ 2 = 2ϕ 3 γ 3 = 3ϕ 4 ϕ 3 γ 4 = 4ϕ 5 3ϕ 4 2 3 ϕ 3 γ j+1 (hj n ) γ 5 = 5ϕ 6 6ϕ 5 11 4 ϕ 4 1 2 ϕ 3 j 1 l l G n,n l=1 12
Convergence assumptions on semilinear initial value problem u (t) = F ( u(t) ) = Au(t) + g ( u(t) ), u(0) = u 0 X Banach space, A : X X e ta X X + t γ A γ e ta X X C γ, γ, t 0. g locally Lipschitz-continuous in a strip along the exact solution f (t) = g ( u(t) ) sufficiently smooth 13
Convergence theorem u (t) = Au(t) + g(u(t)) exponential (k + 1)-step Adams method f (t) = g ( u(t) ) satisfies f C k+1 ([0, T ], X ) Then, if the error bound u j u(t j ) V c 0 h k+1, j = 1,..., k, u n u(t n ) V C h k+1 sup f (k+1) (t) 0 t t n holds uniformly in 0 nh T. C = C(T ), independent of n and h H., Ostermann, 2010 14
Convergence theorem u (t) = Au(t) + g(u(t)) linearized exponential k-step Adams method f (t) = g ( u(t) ) satisfies f C k+1 ([0, T ], X ) Then, if the error bound u j u(t j ) V c 0 h k+1, j = 1,..., k 1, u n u(t n ) V C h k+1 sup 0 t t n ( f (k+1) (t) + u (k+1) (t) V ) holds uniformly in 0 nh T. C = C(T ), independent of n and h H., Ostermann, 2010 14
Outline of proof using interpolation in exact data, exact solution satisfies with defect 1 u(t n+1 ) = e hj n u(t n ) + h e h(1 θ)j n p n (t n + θh) + δ n+1 0 1 δ n+1 = h e h(1 θ)j ( n f n (t n + θh) p n (t n + θh) ) dθ 0 estimating interpolation error yields the bounds f n (t n + θh) p n (t n + θh) δ n+1 Ch k+2, δ n+1 V Ch k+2 α 15
Outline of proof, cont. error recursion 1 e n+1 = e hj n e n + h e h(1 θ)j ( n p n (t n + θh) p n (t n + θh) ) dθ 0 δ n+1 stability is not trivial (H., Ostermann, Schweitzer 2009) employ Lipschitz condition g(u j ) g ( u(t j ) ) L e j V e n V C max e j V + Ch j=1,...,k 1 n 1 j=0 1 t α n j ( ej V + h k+1) stated error bound follows from a discrete Gronwall lemma 16
Numerical example U t 2 U x 2 = 1 + Φ(x, t), x, t [0, 1] 1 + U2 hom. Dirichlet b.c., Φ s.t. U(x, t) = x(1 x) e t, N = 200 grid points error 10 2 10 4 10 6 10 8 10 10 10 12 10 14 h h 2 h 3 h 4 h 5 h 6 10 2 step size 10 1 Adams k = 1,..., 6 linearized Adams k = 1,..., 5 17
Implementation issues error analysis shows that j G n = O(h j ) for sufficiently smooth solutions Krylov subspace methods become cheaper with increasing j (Tokman, 2006) higher order does not cost very much can be exploited by reformulation of linearized exponential Adams methods k 1 u n+1 = u n + hϕ 1 (hj n )F(u n ) + h j=1 γ j+1 (hj n ) j 1 l l G n,n l=1 k 1 = u n + hϕ 1 (hj n )F(u n ) + h β k,l (hj n ) l G n,n l=1 18
Multiple time stepping construction of exponential Adams methods was based on replacing nonlinearity g by local interpolation polynomial p n in v.o.c. formula h u(t n+1 ) = e ha u(t n ) + 0 e (h τ)a g ( u(t n + τ) ) dτ h u n+1 = e ha u n + e (h τ)a p n (t n + τ)dτ 0 different interpretation: u n+1 = y n (h) exact solution of y n(τ) = Ay n (τ) + p n (t n + τ), y n (0) = u n, multiple time stepping: solve this ode by smaller time steps Grote, Mitkova, 2009, 2010 analogously for linearized exponential Adams methods 19
Numerical example, 2d U t U = 1 1 + U 2, x [0, 1]2, t [0, 0.2] hom. Dirichlet b.c., N = 75 grid points in each direction 10 4 j = 1 j = 2 10 5 j = 3 j = 4 10 6 10 7 10 8 0 10 20 30 40 norms of jth backward differences 45 30 20 10 5 3 2 1 0 10 20 30 40 Krylov dimensions j = 0 20
Partitioned problems partitioned semilinear problem [ ] [ ] [ v L Z v = w Y B w ] [ a(t, v, w) + b(t, v, w) ] [ v = A w ] + g(t, v, w) with L corresponding to stiff components, B to nonstiff components [ ] [ ] [ ] [ ] L 0 0 Z v a(t, v, w) A =, g(t, v, w) = + Y 0 0 B w b(t, v, w) matrix functions [ φ( ha) = φ(hl) 0 hy φ [1] (hl) φ(0)i ], φ [1] φ(z) φ(0) (z) = z attractive, if number of stiff components small 21
Application: Simulation of optical resonators optical discs and ring resonators have become essential building blocks of integrated all-optical circuits filtering multiplexing dispersion compensation advanced sensing challenges: long time simulation necessary to extract full spectral behavior of resonator complex geometries joint work with Kurt Busch, Jens Niegemann, KIT Photonics group and Abdullah Demirel 22
1d test problem Maxwell s equation discontinuous Galerkin discretization, order 4 for E x and B z locally refined grid 10 3 p=2 10 3 p=4 10 3 p=6 10 4 10 4 10 4 10 5 0.2 0.3 0.4 0.6 0.8 1 10 5 0.2 0.3 0.4 0.6 0.8 1 10 5 0.2 0.3 0.4 0.6 0.8 1 error versus coarse grid step size, x fine = x coarse /p 23
Mathematical model of ring resonator modeling and spatial discretization by Kurt Busch and Jens Niegemann Maxwell s equation in 2d PML for transparent boundary conditions locally refined grid discontinuous Galerkin discretization 10 d.o.f. per triangle for each of E x, E y, B z total of 38.760 d.o.f. 300 components are considered as stiff small matrix functions computed via diagonalization work in progress... only preliminary results 24
Histogram of radii of triangles incircles 150 100 Count 50 0 0 0.05 0.1 0.15 0.2 0.25 Radius of incircle 25
Eigenvalues 200 100 0 100 200 400 300 200 100 0 26
Spatial grid gain about 30% of computational time against explicit RK method 27
28 Simulation snapshots
Concluding remarks Summary new class of exponential multistep methods convergence analysis of two classes of exponential multistep methods link to multiple time stepping or explicit local time stepping by Mitkova and Grote application to optical resonators Outlook optimize methods for ring resonator simulation different implementations more comparisons with existing methods multiple time stepping rational Krylov methods preconditioning 29