Exponential integrators for oscillatory second-order differential equations

Exponential integrators for oscillatory second-order differential equations Marlis Hochbruck and Volker Grimm Mathematisches Institut Heinrich Heine Universität Düsseldorf Cambridge, March 2007

Outline Motivation Gautschi-type exponential integrators One- and two-step formulations, properties Main result Numerical example: Fermi-Pasta-Ulam problem

Motivation: Sine-Gordon equation u = u xx sinu, t [0, T], x Ω = (0, 1), b.c. + i.c. abstract framework: u = Au g(u), A = u xx, g(u) = sinu u V = D(A 1/2) ) = H0(Ω), 1 u L 2 (Ω) energy: H(u, u ) = 1 2 u 2 + 1 2 A1/2 u 2 + G(u) where T T G(u) = g(tu), u dt = sin(tu(x))u(x)dxdt 0 0 Ω = L 2,, =, L 2

Assumptions gradient operator g: u 2 C m G(u) C M + A 1/2 u 2 with moderate constancs C m, C M finite energy: H e (u, u ) = 1 2 u 2 + 1 2 A1/2 u 2 = H(u, u ) G(u) moderately bounded whenever H(u, u ) is

Some facts A unbouded operator u highly oscillatory in time (temporal derivaties not bounded) u satisfies finite energy assumptions high frequencies cause small time steps energy conservation

Spatial discretization: oscillatory ode second-order ode y + Ω 2 N y = g(y), y(0) = y 0, y (0) = y 0 with finite energy Ω N = A 1/2 N, H(y, y ) = 1 2 y 2 + 1 2 yt Ω 2 N y 1 2 K Ω 2 = A N sym. pos. semidef.

Gautschi-type exponential integrator y + Ω 2 (t, y)y = g(y), y(0) = y 0, y (0) = y 0 for constant g and Ω, exact solution satisfies y(t+h) 2y(t)+y(t h) = h 2 ψ(hω) ( g Ω 2 y(t) ), ψ(x) = sinc 2 x 2 (variation-of-constants formula) Gautschi-type exponential integrator y n+1 2y n + y n 1 = h 2 ψ(hω n ) ( g n Ω 2 ) n y n, Ωn = Ω(t n, y n ) (Verlet for Ω = 0)

Gautschi-type exponential integrator y + Ω 2 (t, y)y = g(y), y(0) = y 0, y (0) = y 0 for constant g and Ω, exact solution satisfies y(t+h) 2y(t)+y(t h) = h 2 ψ(hω) ( g Ω 2 y(t) ), ψ(x) = sinc 2 x 2 (variation-of-constants formula) Gautschi-type exponential integrator y n+1 2y n + y n 1 = h 2 ψ(hω n ) ( g n Ω 2 ) n y n, Ωn = Ω(t n, y n ) (Verlet for Ω = 0) choice of g n?

Gautschi-type exponential integrator y n+1 2y n + y n 1 = h 2 ψ(hω n ) ( g n Ω 2 n y n ) obvious choice: g n = g(y n ) Gautschi 61 resonance problems for hω k jπ, ω k eigenvalue of Ω n better choice: g n = g(φ(hω n )y n ), φ filter function φ(0) = 1, φ(kπ) = 0, k = 1, 2, 3,... convergence result: (H., Lubich, 99, Grimm 02, 05) Assumptions: g smooth, bounded energy: y n y(t n ) h 2 C(t n ), C(t n ) e tnl

Effect of filter function 10 1 without φ with φ 10 2 10 3 10 4 10 5 10 6 10 3 10 2 10 1 10 0

Numerical example: time step h = 0.02 Verlet scheme 128 Fourier modes 512 Fourier modes 2048 Fourier modes 0.05 0.05 0 0 0.05 0 0.5 1 0.05 0 0.5 1 Gautschi-type exponential integrator 0.05 0.05 0.05 0 0 0 0.05 0 0.5 1 0.05 0 0.5 1 0.05 0 0.5 1

One-step formulation rewrite 2nd order ode as system of 1st order odes, apply variation-of-constants formula: exact solution satisfies [ ] y(t + h) y (t + h) =R(hΩ) [ ] y(t) y (t) motivates numerical scheme [ ] [ ] [ yn+1 yn = R(hΩ) + y n+1 y n + t+h t [ Ω 1 ] sin(t + h s)ω g ( y(s) ) ds cos(t + h s)ω 1 2 h2 Ψg(Φy n ) 1 2 h (Ψ 0 g(φy n ) + Ψ 1 g(φy n+1 ) where [ cos hω Ω R(hΩ) := 1 ] sinhω, Φ = φ(hω), Ψ = ψ(hω),... ΩsinhΩ cos hω ] ),

Family of exponential integrators where [ yn+1 y n+1 ] = R(hΩ) [ yn y n ] [ + 1 2 h2 Ψg(Φy n ) 1 2 h (Ψ 0 g(φy n ) + Ψ 1 g(φy n+1 ) Φ = φ(hω), Ψ = ψ(hω), Ψ 0 = ψ 0 (hω), Ψ 1 = ψ 1 (hω) assumptions on ψ, ψ, ψ 0, ψ 1 even analytic functions φ(0) = ψ(0) = ψ 0 (0) = ψ 1 (0) = 1 bounded on non-negative real axis ] ),

Properties of one-step scheme symmetric if and only if ψ(ξ) = sinc(ξ)ψ 1 (ξ), ψ 0 (ξ) = cos(ξ)ψ 1 (ξ), symmetric methods can be cast into equivalent two-step formulation with starting values y n+1 2 cos hω y n + y n 1 = h 2 Ψg(Φy n ), y 0, y 1 = cos hω y 0 + hsinchω y 0 + 1 2 h2 Ψg(Φy 0 ).

One- and two-step formulations, order mollified impulse method (García-Archilla, Sanz-Serna, Skeel, 1998) order 2 as one-step method with particular starting value y0, y 1 as above order 1 as two-step method with exact starting values Gautschi-type integrator (H., Lubich, 1998) order 2 as two-step method for arbitrary starting values close enough to exact solution symmetric one-step formulation leads to ψ1 with singularities at integer multiples of π

Symplecticity necessary and sufficient condition for one-step methods being symplecitic (Hairer, Lubich, Wanner, GNI, 2002) ψ(ξ) = sinc(ξ)φ(ξ), however (Hairer, Lubich, 2000) for Ω = ω > 0 and linear problems, i.e. g(y) = By, energy is conserved up to O(h) for all values of hω if and only if ψ(ξ) = sinc 2 (ξ)φ(ξ)

Symplecticity necessary and sufficient condition for one-step methods being symplecitic (Hairer, Lubich, Wanner, GNI, 2002) ψ(ξ) = sinc(ξ)φ(ξ), however (Hairer, Lubich, 2000) for Ω = ω > 0 and linear problems, i.e. g(y) = By, energy is conserved up to O(h) for all values of hω if and only if ψ(ξ) = sinc 2 (ξ)φ(ξ) indicates that methods satisfying the latter condition are preferable to symplectic ones

Assumptions on filter functions max χ(ξ) M 1, χ = φ, ψ, ψ 0, ψ 1 ξ 0 max φ(ξ) 1 ξ 0 ξ M 2. ( 1 max ξ 0 sin ξ sinc 2ξ ) 2 ψ(ξ) M 3 2 1 max ξ 0 ξ sin ξ (sincξ χ(ξ)) M 4, χ = φ, ψ 0, ψ 1 2 [ yn+1 y n+1 ] = R(hΩ) [ yn y n ] [ + 1 2 h2 Ψg(Φy n ) 1 2 h (Ψ 0 g(φy n ) + Ψ 1 g(φy n+1 ) ] ),

Theorem (Grimm, H., 2006) Assumptions; exact solution y satisfies finite-energy condition conditions on filter functions are satisfied then y(t n ) y n h 2 C, t 0 t n = t 0 + nh t 0 + T where the constant C depends on T, K, M 1,...,M 4, g, g y, and g yy. additional conditions on filter functions y (t n ) y n hc, t 0 t n = t 0 + nh t 0 + T

Numerical examples Fermi-Pasta-Ulam problem stiff harmonic soft nonlinear new choice of filter function (H., Grimm, 06) ψ(ξ) = sinc 3 (ξ), φ(ξ) = sinc(ξ) order two energy conserved up to O(h) for linear problems

Gautschi-type methods ψ(ξ) φ(ξ) A sinc 2 ( 1 2ξ) 1 Gautschi, 61 B sinc(ξ) 1 Deuflhard, 79 C sinc(ξ)φ(ξ) sinc(ξ) García-Archilla et al., 96 D sinc 2 ( 1 2 ξ) sinc(ξ)( 1 + 1 3 sin2( 1 2 ξ)) H., Lubich, 98 E sinc 2 (ξ) 1 Hairer, Lubich, 00 G sinc 3 (ξ) sinc(ξ) Grimm, H., 06

Maximum error of total energy (A) (B) 0.2 0.2 0.2 (C) 0.1 0.1 0.1 π 2π 3π 4π π 2π 3π 4π π 2π 3π 4π 0.2 (D) 0.2 (E) 0.2 (G) 0.1 0.1 0.1 π 2π 3π 4π π 2π 3π 4π π 2π 3π 4π interval [0, 1000], h = 0.02, error vs. hω

Global error at t = 1 10 0 (A) 10 0 (B) 10 0 (C) 10 2 10 2 10 2 10 4 10 4 10 4 10 6 10 2 10 1 10 6 10 2 10 1 10 6 10 2 10 1 10 0 (D) 10 0 (E) 10 0 (G) 10 2 10 2 10 2 10 4 10 4 10 4 10 6 10 2 10 1 10 6 10 2 10 1 10 6 10 2 10 1 error vs. step size, ω = 1000

Maximum deviation of oscillatory energy (A) (B) (C) 0.2 0.2 0.2 0.1 0.1 0.1 π 2π 3π 4π π 2π 3π 4π π 2π 3π 4π 0.2 (D) 0.2 (E) 0.2 (G) 0.1 0.1 0.1 π 2π 3π 4π π 2π 3π 4π π 2π 3π 4π interval [0, 1000], h = 0.02, error vs. hω

Comments on new method order 2 (according to theorem) nearly conserved energy (for linear problems according to Hairer, Lubich, 00) no resonances for oscillatory energy (surprise, because there is no method which uniformely conserves oscillatory energy on interval of length > 2π for linear problems, Hairer, Lubich 00)

Sketch of proof substitute exact solution into numerical scheme defects derive expressions for defects substract numerical solution, obtain error recursion (discrete variation-of-constants formula) use explicit expression for defects to bound all sums arising apply Gronwall Lemma

Summary nonsmooth error bounds for family of exponential integrators characterized second order methods in terms of properties of filter functions accuracy in time independent on spatial discretization results valid for abstract ode s suggest new choice of filter function with favorable properties on fpu example