High-order time-splitting methods for Schrödinger equations

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1 High-order time-splitting methods for Schrödinger equations and M. Thalhammer Department of Mathematics, University of Innsbruck Conference on Scientific Computing 2009 Geneva, Switzerland

Nonlinear Schrödinger equations Bose Einstein condensation In our laboratories temperatures are measured in micro- or nanokelvin... In this ultracold world... atoms move at a snail s pace.

2 Nonlinear Schrödinger equations Bose Einstein condensation In our laboratories temperatures are measured in micro- or nanokelvin... In this ultracold world... atoms move at a snail s pace... and behave like matter waves. Interesting and fascinating new states of quantum matter are formed and investigated in our experiments. Grimm et al. Practical realisation. Observation of Bose Einstein condensation in physical experiments. Theoretical model. Mathematical description by systems of nonlinear Schrödinger equations. Numerical simulation. Favourable discretisations rely on time-splitting pseudospectral methods. See recent work by Bao, Cancès, Dion, Du, Jaksch, Markowich, Pérez-García, Shen, Tang, Tiwari, Shukla, Vazquez, Zhang etc.

3 Objectives Nonlinear Schrödinger equations Gross Pitaevskii equation (GPE). Nonlinear Schrödinger equation of the form i t ψ(x, t) = ( 2 2 m + U(x) + g ψ(x, t) 2) ψ(x, t). Numerical solution. Aims. Space discretisation based on pseudospectral methods. Time discretisation based on exponential operator splitting methods. Convergence analysis for linear Schrödinger equations. Comparison of high-order splitting methods regarding accuracy, efficiency and conservation of geometric properties (particle number, energy).

4 Outline Nonlinear Schrödinger equations 1 Nonlinear Schrödinger equations Gross Pitaevskii equation Abstract formulation 2 Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity 3 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE

5 Gross Pitaevskii equation Gross Pitaevskii equation Abstract formulation Gross Pitaevskii equation (GPE). Normalised nonlinear Schrödinger equation i t ψ(x, t) = ( V (x) + ϑ ψ(x, t) 2) ψ(x, t). describes wave function of Bose Einstein condensate. Harmonic trap. Consider physically relevant case of harmonic potential V (x) = 1 2 V H(x) = 1 2 dj=1 γ 2 j x 2 j Geometric properties. Conservation of particle number ψ(, t) L t x 2 1 and energy E ( ψ(, t) ) ( ( = V ϑ ψ(, t) 2) ) ψ(, t) ψ(, t) L 2 3

6 Gross Pitaevskii equation Gross Pitaevskii equation Abstract formulation Gross Pitaevskii equation (GPE). Nonlinear Schrödinger equation i t ψ(x, t) = ( V (x) + ϑ ψ(x, t) 2) ψ(x, t). describes the wave function ψ : R d [0, ) C of a Bose Einstein condensate. Groundstate solution. Special solution of the form ψ(x, t) = e iµt φ(x) that minimizes energy functional. Minimisation approach. Compute ground state by direct minimisation of energy functional, see Bao and Tang (2003) and Caliari et al. (2009). Use ground state as reliable reference solution for time integration.

7 Evolutionary Schrödinger equations Gross Pitaevskii equation Abstract formulation Abstract formulation. Interpret time-dependent nonlinear Schrödinger equation as abstract evolution equation u (t) = ( A + B(u(t)) ) u(t). Numerical solution. Split right hand side into two parts Hermite Fourier A = i 1 ( ) 2 VH A = i 1 2 B(u) = i ϑ u 2 B(u) = i ( 1 2 V H + ϑ u 2 ) Numerical method relies on solutions of evolution equations v (t) = A v(t) v(0) = v 0 given w (t) = B(w(t)) w(t) w(0) = w 0 given

8 Hermite spectral method Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity Spectral decomposition of harmonic oscillator. Properties of the harmonic oscillator: Hermite functions (H µ ) form ON-basis of L 2 (R d ) v = v µ H µ, v µ = (v H µ ) L 2. µ N d Hermite functions are eigenfunctions of i A = 1 ( ) 2 + VH ( ) VH Hµ = λ µ H µ, λ µ = d j=1 γ j ( µ j). Exact solution of v (t) = A v(t), v(0) = v µ H µ is given by v(t) = µ N d e itλµ v µ H µ.

9 Hermite pseudospectral method Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity Numerical approximation. Truncate infinite sum and use the first M Hermite functions v(t) = v µ (t)h µ. µ M 1 Computation of spectral coefficients. Approximation of Hermite spectral coefficients by Gauss Hermite quadrature formula v µ = v(x) H µ (x) dx ω µ e ξ k 2 v(ξ k ) H µ (ξ k ). R d k Implementation. Hermite transform and inverse Hermite transform in 2D can be realised by two matrix matrix multiplications. Complexity of Hermite transform in 2D is O(M 3 ).

10 Fourier spectral method Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity Spectral decomposition of Laplacian. Properties of the Laplacian: Fourier basis functions (F µ ) form ON-basis of L 2 (( a, a) d ) v = µ Z d v µ F µ, v µ = (v F µ ) L 2. Fourier basis functions are eigenfunctions of i A = F µ = λ µ F µ, λ µ = 1 2 a 2 π 2 d µ 2 j. j=1 Exact solution of v (t) = A v(t), v(0) = v µ F µ is given by v(t) = µ Z d e itλµ v µ F µ.

11 Fourier pseudospectral method Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity Numerical approximation. Truncate infinite sum and use the first M Fourier functions v(t) = M 2 v µ (t)h µ. µ M 2 1 Computation of spectral coefficients. Approximation of Fourier spectral coefficients by trapezoidal rule with equidistant grid points v µ = v(x) F µ (x) dx ω v(ξ k ) F µ (ξ k ). ( a,a) d Implementation. Use Fast Fourier Transformation (FFT). Complexity of FFT in 2D is O(log(M) M 2 ).

12 Numerical experiment (CPU-time) Computation time of spectral methods. Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity 10 2 Hermite 1D Fourier 1D Hermite 2D Fourier 2D CPU seconds CPU seconds degree of freedom degree of freedom Figure: Computation time of the Hermite and Fourier spectral methods in one (left picture) and two (right picture) space dimensions using M = 2 i, 4 i 8, basis functions in each space direction.

13 Numerical experiment (spatial error) Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity error ϑ = 1 ϑ = 10 ϑ = 100 ϑ = degree of freedom error ϑ = 1 ϑ = 10 ϑ = 100 ϑ = degree of freedom Figure: Spatial error of the Hermite (left picture) and Fourier (right picture) spectral method for different values of the coupling constant ϑ.

14 Pseudospectral methods Hermite vs. Fourier Numerical solution of nonlinearity Numerical approximation of nonlinear part B Exact solution. Exact solution of the nonlinear part ( w (t) = i ɛv + ϑ w(t) 2) w(t), ɛ {0, 1}, is given by ( ( w(t) )(x) = e it ɛv (x)+ϑ w 0 (x) 2) w 0 (x). Numerical approximation. Evaluate pointwise multiplication only on the Gauss-Hermite quadrature nodes ξ k. Implementation. Numerical approximation in 2D can be realised by a pointwise matrix-matrix multiplication. Complexity in 2D is O(M 2 ).

15 Exponential operator splitting methods Problem class. Linear evolution equation Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE u (t) = A u(t) + B u(t), t 0, u(0) given. Compute numerical approximation u n u(t n ) at time t n = h n. Example method. Strang or symmetric Lie Trotter splitting u n+1 = e h 2 A e hb e h 2 A u n, see Trotter (1959) and Strang (1968). Method class. Higher-order exponential operator splitting methods of the form s u n+1 = e bjha e ajhb u n. j=1 with coefficients a j, b j (1 j s).

16 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Splitting methods for nonlinear Schrödinger equations Problem class. Nonlinear evolution equation of the form u (t) = A u(t) + B(u(t)) u(t), t 0, u(0) given. Example method. Strang or symmetric Lie Trotter splitting u n+1 = e h 2 A e hb(e h 2 A u n) e h 2 A u n. Method class. Higher-order exponential operator splitting methods of the form U n,1 = e a 1hA u n 1, U n,j = e a jha e b j 1hB(U n,j 1 ) U n,j 1, 2 j s, u n = e bshb(un,s) U n,s, with coefficients a j, b j (1 j s).

17 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE High-order exponential operator splitting methods method order #Compositions McLachlan McLachlan p = 2 s = 3 Strang Strang p = 2 s = 2 BM4-1 Blanes & Moan PRKS 6 p = 4 s = 7 BM4-2 Blanes & Moan SRKN b 6 p = 4 s = 7 M4 McLachlan p = 4 s = 6 S4 Suzuki p = 4 s = 6 Y4 Yoshida p = 4 s = 4 BM6-1 Blanes & Moan PRKS 10 p = 6 s = 11 BM6-2 Blanes & Moan SRKN b 11 p = 6 s = 12 BM6-3 Blanes & Moan SRKN a 14 p = 6 s = 15 KL6 Kahan & Li p = 6 s = 10 S6 Suzuki p = 6 s = 26 Y6 Yoshida p = 6 s = 8 Table: Splitting methods of order p involving s compositions.

18 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Schrödinger equation involving unbounded potential Problem class. Linear Schrödinger equation of the form i t ψ(x, t) = ( V (x) + W (x)) ψ(x, t) involving unbounded polynomial potential W (x) = α µ x µ, α µ R. µ N d. Consider abstract evolution equation and split right hand side into A v(t) = i 1 2 ( V H(x)) v(t) and B w(t) = i W (x) w(t). Aim. Convergence result for exponential operator splitting Hermite pseudospectral methods of arbitrary order.

19 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Convergence result for linear Schrödinger equation Theorem (Neuhauser and Thalhammer (2009)) Suppose that the coefficients of the splitting method satisfy the classical order conditions for p 1. Then, provided that the exact solution u(t) is sufficiently regular, the following estimate holds u n u(t n ) L 2 C u 0 u(0) L 2 + C h p, 0 nh T Y KL S BM6 1 BM BM Strang McLachlan Y S M BM4 1 BM

20 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Numerical experiment (long-term integration) Numerical experiment (Caliari et al. (2009)). Illustrates the temporal order of Hermite and Fourier pseudospectral splitting methods for the GPE i t ψ(x, t) = ( V (x) + ψ(x, t) 2) ψ(x, t) involving the potential V (x) = 1 2 (x x 2 2). Choose the ground state solution of the GPE 0.7 involving the potential V (x) = x1 2 + x as initial value. Compute a reference solution at T = 400 ( basis functions, 2 17 time steps) t x

21 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Numerical experiment (long-term integration) Numerical experiment. Illustrates the temporal order of Hermite and Fourier pseudospectral splitting methods for the GPE i t ψ(x, t) = ( V (x) + ψ(x, t) 2) ψ(x, t). Compute numerical approximations by time-splitting spectral methods with 2 i, 6 i 8, basis functions in each space direction and 2 i, 6 i 15, time steps. Compute numerical approximations by standard explicit Runge Kutta methods of order four. Compute error in discrete L 2 -norm. Measure the particle number and energy conservation.

22 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Numerical experiment (long-term integration) tol. method d.o.f. #transf. pn E < 10 2 Hermite < 10 2 Fourier < 10 2 Hermite < 10 2 Fourier < 10 2 Hermite < 10 2 Fourier < 10 2 Hermite rk < 10 2 Fourier rk < 10 2 Hermite ode < 10 2 Fourier ode

23 Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Numerical experiment (long-term integration) tol. method d.o.f. #transf. pn E < 10 4 Hermite < 10 4 Fourier < 10 4 Hermite < 10 4 Fourier < 10 4 Hermite rk < 10 4 Fourier rk < 10 4 Hermite ode < 10 4 Fourier ode

24 Observations Nonlinear Schrödinger equations Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE The number of required Hermite basis functions is smaller than the number of Fourier basis functions. The number of required Hermite transforms is (in general) smaller than the required Fourier transforms. The splitting methods outperform the explicit Runge Kutta methods. The splitting methods of order four and six are superior to the second order Strang splitting.

25 Conclusion and future work Exponential operator splitting Convergence analysis of linear Schrödinger equations Long-term integration of GPE Contents. High accuracy discretisations by time-splitting pseudospectral methods. Convergence analysis for linear evolutionary Schrödinger equations. Comparison of time-splitting methods regarding accuracy, efficiency, and geometric properties. Future work. Provide convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. Study other basis functions, well adapted for a certain potential.

Three approaches for the design of adaptive time-splitting methods

Three approaches for the design of adaptive time-splitting methods Mechthild Thalhammer Leopold Franzens Universität Innsbruck, Austria Workshop on Geometric Integration and Computational Mechanics Organisers: