Efficient Multisymplectic Algorithms For Dirac Equations

This paper studies the split-step multi-symplectic algorithm for nonlinear Dirac equation.Multi-symplectic split-step algorithm is comby the split-step algorithm and multi-symplectic algorithm to study the nonlinear Dirac equation. The basic idea is to split nonlinear Dirac equation into linear and nonlinear sub-problems with symplectic or multi-symplect structure, and then respectively with symplectic scheme is used to discrete them, get the format with integral sympletic,which explain the universality and e?ectiveness of the split-step multi-symplectic algorithms.In the first chapter, we mainly introduce the research background of some numerical methods for the nonlinear Dirac equation and have been applied to the solution of the Dirac equation, and then gives some basics knowledge of this article need to know.In the second chapter, we begin a detailed study of the multi-symplectic algorithm for nonlinear Dirac equation. First constructed multi-symplectic structural of the nonlinear Dirac equation,and then use the Runge-Kutta format to discrete and numerical simulation. Next applying splitstep algorithm to the Dirac equation, construct a split-step multi-symplectic algorithm for nonlinear Dirac equation, and then using di?erent formats discrete. The first discrete method is use the midpoint of discrete format discrete; The second discrete method is use higher-order compact format to discrete in space, time with symplectic Euler method of discrete, and analyze the stability of conditional of scheme. Finally, through numerical experiments compare with the traditional multi-symplectic algorithms, reflecting the advantages of split-step multi-symplectic algorithms for further analysis.In the third chapter the split-step multi-symplectic algorithms is extended to nonlinear twodimensional nonlinear Dirac equation, analysis the stability of condition, the final numerical experiments, analyze the feasibility of their approach.In the fourth chapter, a brief summary of the content of this paper. then puts forward the tentative idea of further study on nonlinear Dirac equation.