Symplectic And Multisymplectic Structure-preserving Numerical Schemes For Evolution Equation

This paper studies the symplectic Euler scheme for the one-dimensional high order Schr(o|¨)dinger equa-tion and the split-step multi-symplectic scheme for the two-dimensional nonlinear Schr(o|¨)dinger equation.To discretize the temporal direction for the semi-discrete Hamiltonian system, applying backwardEuler scheme to discretize the first equation and using the forward Euler method to the second equation,which is called symplectic Euler scheme, namely first stage and first order Runge-Kutta scheme. Thisscheme seems to be implicit, but it is essentially explicit in the process of implementation, which makethe symplectic Euler scheme not only symplectic structure-preserving, but also needn't resolve couplednonlinear algebraic equations, moreover, this scheme runs faster and has higher accuracy compared to thegeneral Euler method. The specific analysis and discussion about the detailed behavior of the symplecticEuler scheme is in chapter 2, including the conservation, stability and error estimation. In addition, toreflect the superiority of the symplectic Euler scheme, the paper compares this scheme to the backwardEuler method.Split-step multi-symplectic method is to discretize the Hamiltonian system by combining the split-stepalgorithm and multi-symplectic algorithm, whose basic idea is to split the original complex system into anumber of simpler subsystems, then use multi-symplectic scheme to discretize these subsystems respec-tively. Finally, we can achieve the numerical simulation solution of the original system according to a certaincombination sequential of the subsystems. This method can not only conserve multi-symplectic geometrystructure of the original system, conduct numerical simulation over long time, but also have advantages of?exibility and easy to modular. Therefore, the split-step multi-symplectic scheme is more suitable to solvecomplex problems and multi-dimensional problems compared to multi-symplectic scheme. In chapter 3,we give the concrete construction method and analysis of the split-step multi-symplectic scheme, and fullyillustrate the economics and e?ectiveness of the split-step multi-symplectic scheme through comparison tomulti-symplectic scheme.