| In order to use structure-preserving algorithms (SPA) to simulate partial differ-ential equations, Y. S. Wang et al. proposed the concept of local structure-preserving algorithm (LSPA). Multi-symplectic algorithm (MSA) is a natural extension to the symplectic algorithm for Hamiltonian ordinary differential equations, and also be-longs to the LSPA. How to construct the MSA systematically is a fundamental prob-lem in the development of the theory of MSA. In this thesis, starting from the La-grangian mechanic and basing on the discrete variational principle, we derive several variational integrators for a nonlinear Schrodinger equation by discretizing the cor-responding Lagrangian functional. These derived integrators can automatically con-serve the discrete multi-symplectic conservation laws, which make these integrators to be multi-symplectic integrators. A series of numerical experiments are also presented in this thesis to show the efficiency of the MSA to simulate the nonlinear Schrodinger equation.Firstly, we give a brief introduction to the multi-symplectic geometry, the to-tal variational principle and the variational integrators. And we derive the multi-symplectic structure of the nonlinear Schrodinger by its total variational principle and obtain the corresponding multi-symplectic conservation law. Secondly, based on the discrete variational principle, we derive several variational integrators for the nonlin-ear Schrodinger equation by using the triangular and rectangular grids to discrete the base space and applying different difference schemes to discretize the corresponding Lagrangian functional. We also show that one of the new-derived integrators is es-sentially equivalent to the well-known six-point multi-symplectic scheme. Finally, we verify the efficiency of the multi-symplectic schemes by various numerical exper-iments. Numerical results also show the excellent performance of MSA in long-time behavior and invariants-preserving properties. |
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