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.