zbMATH — the first resource for mathematics

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. (English) Zbl 1198.65186
Summary: 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.

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] G.P. Agrawal, Nonlinear fiber optics, Fourth edition, Elsevier Books, Oxford, 2006.
[2] 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 , NIC Series Vol. 10, John von Neumann Institute for Computing, Jülich (2002), 255-268.
[3] 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. · Zbl 1028.82501
[4] Weizhu Bao, N. J. Mauser, and H. P. Stimming, Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-\?\? model, Commun. Math. Sci. 1 (2003), no. 4, 809 – 828. · Zbl 1160.81497
[5] 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. · Zbl 1026.65073
[6] 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. · Zbl 0739.35080
[7] F. Castella, \?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. · Zbl 0892.35141
[8] M. Fröhlich, Exponentielle Integrationsverfahren für die Schrödinger-Poisson-Gleichung, Doctoral Thesis, Univ. Tübingen, 2004.
[9] 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. · Zbl 1094.65125
[10] 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. · Zbl 0789.65048
[11] 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.
[12] 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. · Zbl 1030.65100
[13] 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. · Zbl 0808.35116
[14] Tobias Jahnke and Christian Lubich, Error bounds for exponential operator splittings, BIT 40 (2000), no. 4, 735 – 744. · Zbl 0972.65061
[15] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009
[16] 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. · Zbl 1053.65061
[17] 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. · Zbl 1037.81634
[18] Robert I. McLachlan and G. Reinout W. Quispel, Splitting methods, Acta Numer. 11 (2002), 341 – 434. · Zbl 1105.65341
[19] Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506 – 517. · Zbl 0184.38503
[20] 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. · Zbl 0928.35157
[21] 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. · Zbl 0597.76012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.