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Zbl 1110.65116
Hong, Jialin; Liu, Ying; Munthe-Kaas, Hans; Zanna, Antonella
Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients. (English)
[J] Appl. Numer. Math. 56, No. 6, 814-843 (2006). ISSN 0168-9274

Authors' summary: Based on the multi-symplecticity of the Schrödinger 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, comparisons with the 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.
[Rudolf Scherer (Karlsruhe)]
MSC 2000:
*65P10 Hamiltonian systems including symplectic integrators
37K10 Completely integrable systems etc.
37M15 Symplectic integrators
35Q55 NLS-like (nonlinear Schroedinger) equations
65M15 Error bounds (IVP of PDE)
65M06 Finite difference methods (IVP of PDE)

Keywords: consevation laws; error estimation; global energy transit; multi-symplectic integrators; comparison of methods; numerical experiments; Goldberg scheme; Crank-Nicolson scheme

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