On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

Lubich, Christian

doi:10.1090/S0025-5718-08-02101-7

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
HTML articles powered by AMS MathViewer

by Christian Lubich HTML | PDF

Math. Comp. 77 (2008), 2141-2153 Request permission

Abstract:

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an $H^4$-regular solution, a first-order error bound in the $H^1$ norm is shown and used to derive a second-order error bound in the $L_2$ norm. For the cubic Schrödinger equation with an $H^4$-regular solution, first-order convergence in the $H^2$ norm is used to obtain second-order convergence in the $L_2$ norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and $H^m$-conditional stability for error propagation, where $m=1$ for the Schrödinger-Poisson system and $m=2$ for the cubic Schrödinger equation.

References

G.P. Agrawal, Nonlinear fiber optics, Fourth edition, Elsevier Books, Oxford, 2006.
H. Appel, E.K.U. Gross, Static and time-dependent many-body effects via density-functional theory, in Quantum simulations of complex many-body systems: From theory to algorithms (J. Grotendorst, D. Marx, A. Muramatsu, eds.), NIC Series Vol. 10, John von Neumann Institute for Computing, Jülich (2002), 255–268.
Weizhu Bao, Dieter Jaksch, and Peter A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), no. 1, 318–342. MR 1977789, DOI 10.1016/S0021-9991(03)00102-5
Weizhu Bao, N. J. Mauser, and H. P. Stimming, Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-$\rm X\alpha$ model, Commun. Math. Sci. 1 (2003), no. 4, 809–828. MR 2041458
Christophe Besse, Brigitte Bidégaray, and Stéphane Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40 (2002), no. 1, 26–40. MR 1921908, DOI 10.1137/S0036142900381497
Franco Brezzi and Peter A. Markowich, The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation, Math. Methods Appl. Sci. 14 (1991), no. 1, 35–61. MR 1087449, DOI 10.1002/mma.1670140103
F. Castella, $L^2$ solutions to the Schrödinger-Poisson system: existence, uniqueness, time behaviour, and smoothing effects, Math. Models Methods Appl. Sci. 7 (1997), no. 8, 1051–1083. MR 1487521, DOI 10.1142/S0218202597000530
M. Fröhlich, Exponentielle Integrationsverfahren für die Schrödinger-Poisson-Gleichung, Doctoral Thesis, Univ. Tübingen, 2004.
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR 1227985
R.H. Hardin, F.D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Review 15 (1973), 423.
Willem Hundsdorfer and Jan Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Series in Computational Mathematics, vol. 33, Springer-Verlag, Berlin, 2003. MR 2002152, DOI 10.1007/978-3-662-09017-6
Reinhard Illner, Paul F. Zweifel, and Horst Lange, Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci. 17 (1994), no. 5, 349–376. MR 1273317, DOI 10.1002/mma.1670170504
Tobias Jahnke and Christian Lubich, Error bounds for exponential operator splittings, BIT 40 (2000), no. 4, 735–744. MR 1799313, DOI 10.1023/A:1022396519656
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
Roman Kozlov, Anne Kværnø, and Brynjulf Owren, The behaviour of the local error in splitting methods applied to stiff problems, J. Comput. Phys. 195 (2004), no. 2, 576–593. MR 2046751, DOI 10.1016/j.jcp.2003.10.011
Christian Lubich, A variational splitting integrator for quantum molecular dynamics, Appl. Numer. Math. 48 (2004), no. 3-4, 355–368. Workshop on Innovative Time Integrators for PDEs. MR 2056923, DOI 10.1016/j.apnum.2003.09.001
Robert I. McLachlan and G. Reinout W. Quispel, Splitting methods, Acta Numer. 11 (2002), 341–434. MR 2009376, DOI 10.1017/S0962492902000053
Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI 10.1137/0705041
Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23 (1986), no. 3, 485–507. MR 842641, DOI 10.1137/0723033