Exponential integrators

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1 Exponential integrators Marlis Hochbruck Heinrich-Heine University Düsseldorf, Germany Alexander Ostermann University Innsbruck, Austria Helsinki, May 25 p.1

2 Outline Time dependent Schrödinger equations application: quantum dynamics methods: Magnus integrators 2nd order differential equations application: laser/plasma interaction nonlinear wave equations methods: Gautschi-type integrators p.2

3 Quantum dynamics simulations Schrödinger equation: ψ (t) = ih(t)ψ(t), H(t) = U + V (t), U x (hence H(t) large, 1/ x 2 ) standard scheme: split-step Fourier method p.3

4 Quantum dynamics simulations Schrödinger equation: ψ (t) = ih(t)ψ(t), H(t) = U + V (t), U x (hence H(t) large, 1/ x 2 ) standard scheme: split-step Fourier method alternative: exponential midpoint rule (Magnus method) ψ n+1 = exp ( ihh(t n+1/2 ) ) ψ n (exact for constant H) p.3

5 Magnus integrators y = A(t)y(t), y() = y Magnus, 1954: determine Ω(t) such that y(t) = exp(ω(t))y Magnus expansion (valid for Ω(t) < π) Ω(t) = t A(τ)dτ t t t τ [ [ τ [ τ A(σ)dσ, A(τ)] dτ σ [ A(µ)dµ, A(σ)] dσ, A(τ)] dτ A(σ)dσ, [ τ A(µ)dµ, A(τ)]] dτ +... p.4

6 Numerical methods Magnus integrators y n+1 = exp(ω n )y n, Ω n Ω(h) (review: Iserles, Munthe-Kaas, Nørsett, Zanna, ) Approximation involves truncating the Magnus expansion (after k terms) t k = 1 : k = 2 : Ω(t) Ω(t) t A(t n + τ)dτ A(t n + τ)dτ 1 2 t τ [ A(t n + σ)dσ, A(t n + τ)] dτ approximating integrals by replacing A(t) by interpolation polynomial Â(t) for quadrature nodes t n + c j h p.5

7 Examples of Magnus integrators k = 1, exponential midpoint rule Ω n = ha(t n + h/2). k = 2, two-point Gauß quadrature rule: Ω n = h 2 (A 1 + A 2 ) + 3h 2 12 [A 2, A 1 ], A j = A(t n + c j h), c j nodes of Gauß quadrature rule k = 2, method by Blanes, Casas, Ros Ω n = h 6 ( A(tn )+4A(t n+1/2 )+A(t n+1 ) ) h2 12 [A(t n), A(t n+1 )]. p.6

8 Convergence for Schrödinger equations Theorem (H., Lubich, 23) Exponential midpoint rule y n y(t n ) Ch 2 t n 4th-order Gauß method: for h D c y n y(t n ) Ch 4 t n max Dy(t) t t n max D 3 y(t) t t n p.7

9 Convergence for Schrödinger equations Theorem (H., Lubich, 23) Exponential midpoint rule y n y(t n ) Ch 2 t n 4th-order Gauß method: for h D c y n y(t n ) Ch 4 t n max Dy(t) t t n max D 3 y(t) t t n error bound for classical implicit midpoint rule: y n y(t n ) Ch 2 t n max d3 t t n dt 3 y(t) p.7

10 Numerical experiment i ψ t = 1 2 ψ + b(x, t)ψ, x = (x 1,..., x d ) R d, t > discretized s. t. h D 3.5 (N = 32,..., 248 Fourier modes) midpoint Gauss 4 Blanes et al time steps versus error p.8

11 2nd order differential equations y = f(y), y() = y, y () = y standard scheme: leap-frog, Störmer-Verlet y n+1 2y n + y n 1 = h 2 f(y n ) nice properties: order two, symmetric, symplectic Problem: stability requires hω max 2 (1/2) derive alternative without this restriction for y + Ω 2 (t, y)y = g(y) which reduces to leap-frog for Ω = p.9

12 Application: Laser plasma interaction a A =.1, Q =.3 density profile pulses at different times z/λ a A =.12, Q =.3 density profile pulses at different times z/λ p.1

13 Physical model a vectorpotential, E parallel component of electrical field 2 E t 2 2 a z 2 2 a t 2 = Q (n + δn) γ + Q n E = n γ z a where n plasma profile and γ 2 = 1 + a 2 δn = E z relativistic factor density variation p.11

14 2nd order differential equations y + Ω 2 (t, y)y = g(y), y() = y, y () = y 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 ) choice of g n? p.12

15 2nd order differential equations 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 p.13

16 2nd order differential equations 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() = 1, φ(kπ) =, k = 1, 2, 3,... p.13

17 2nd order differential equations 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() = 1, φ(kπ) =, k = 1, 2, 3,... convergence result: (H., Lubich, 99, Grimm 2, 5) Assumptions: g smooth, bounded energy: y n y(t n ) h 2 C(t n ), C(t n ) e t nl p.13

18 Effect of filter function 1 1 without φ with φ p.14

19 Generalization: multiple time-stepping y = f(y) + g(y) given y n and y n compute suitable averaged value y n and solution of u = f(u) + g(y n ), u() = y n, u () = y n. (e.g. by numerical methods with smaller time steps) compute y n+1 and y n+1 from y n+1 2y n + y n 1 = u(h) 2u() + u( h), y n+1 y n 1 = u (h) u ( h), symmetric multiple-time-stepping scheme p.15

20 Results for laser / plasma simulation maximum amplitude error of 1% (phase error ignored) A =.12, Q =.3 dz dt runtime save leap frog leap frog + quasi-envelope % Gautschi % Gautschi + quasi-envelope % PIC multilevel approach not included! 1 day Karle, Schweitzer, H., Laedke, Spatschek, Preprint 25 p.16

21 Quasi-envelope approach x 1 3 A =.12, Q =.3, spectrum k/k replace vectorpotential a by a(z, t) = ã(z, t)e iκz p.17

22 Phase errors at t = t end intensity A =.12, Q =.3 ref PIC KG all tricks z/λ p.18

23 Possible topics for informal sessions Basics on numerical time integration Implementation of exponential integrators (Krylov subspace methods) Numerical methods for time-dependent Schrödinger equations (Magnus integrators) Numerical methods for nonlinear wave equations (Laser / plasma interaction) Numerical methods for parabolic problems p.19

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