Some New Structure-preserving Algorithms For The Multi-symplectic Hamiltonian System

Many partial differential equations (PDEs), for instance the sine-Gordon equa-tion, the nonlinear Schrodinger equation, the Korteweg-de Vries equation, the Camassa-Holm equation, the Maxwell’s equations, the nonlinear wave equation and so on, can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conserva-tion law and local momentum conservation law. It is very significant to construct the numerical methods that preserve some of the above conservation laws.As the multi-symplectic conservation law is an important geometric property of the multi-symplectic Hamiltonian system, a large number of numerical integrators, which preserve discrete multi-symplectic conservation law, have been developed in the past two decades. In this dissertation, we investigate the multi-symplectic Fourier pseudospectral method for the Kawahara equation, and establish the relationship be-tween spectral differentiation matrix and discrete Fourier transform. Due to the rela-tionship, we can apply the fast Fourier transform to solve the discrete system.The energy conservation is a crucial property of mechanical systems and plays an important role in the study of properties of solutions. In some examples, stabil-ity of a numerical method is proved by directly using energy conservative property. Because energy is the most important first integral of many evolutional equations, energy-preserving algorithms have been naturally interesting to researchers and there-fore developed very fast. In this dissertation, by applying the wavelet collocation method in space and the average vector field method in time, we construct a system-atic method for general multi-symplectic Hamiltonian system, while conserving their global energy exactly. We also present a local energy-preserving algorithm for general multi-symplectic Hamiltonian system.Besides energy conservation law, multi-symplectic Hamiltonian system admits momentum conservation law which is also an important invariant in physics. But there are few momentum-preserving methods in literatures. In this dissertation, we give a local momentum-preserving method for general multi-symplectic Hamiltonian system. It is worth noting that the local energy-preserving method and the local momentum-preserving method are independent of the boundary conditions and can be used for a huge class of conservative PDEs.In this dissertation, we specially construct a conservative Fourier pseudospectral algorithm for the coupled nonlinear Schrodinger equations. We prove an important result that semi-norm induced by the Fourier pseudospectral method is equivalent to that of the finite difference method. Due to this result and the fact that the proposed scheme conserves the discrete total mass and energy, the Fourier pseudospectral so-lution is proved to be bounded in the discrete L∞ norm. Then, the scheme is showed to be uniquely solvable and unconditionally stable. The error estimate in the discrete L2 norm is established without assumptions beyond those necessary for existence and uniqueness of the differential equation. Numerical experiments are presented to sup-port the theoretical analysis.