Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients

Authors:
Jialin Hong;Ying Liu;Hans Munthe-Kaas;Antonella Zanna
Affiliations:
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Acade ...;Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China;Institutt for Informatikk, N-5020 Bergen, Norway;Institutt for Informatikk, N-5020 Bergen, Norway
Venue:
Applied Numerical Mathematics
Year:
2006

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Abstract

Based on the multi-symplecticity of the Schrodinger equations with variable coefficients, we give a multi-symplectic numerical scheme, and investigate some conservative properties and error estimation of it. We show that the scheme satisfies discrete normal conservation law corresponding to one possessed by the original equation, and propose global energy transit formulae in temporal direction. We also discuss some discrete properties corresponding to energy conservation laws of the original equations. In numerical experiments, the comparisons with modified Goldberg scheme and Modified Crank-Nicolson scheme are given to illustrate some properties of the multi-symplectic scheme in the numerical implementation, and the global energy transit is monitored due to the scheme does not preserve energy conservation law. Our numerical experiments show the match between theoretical and corresponding numerical results.