Multi-symplectic Methods For Some Nonlinear Wave Equations

Manyimportantmathematicalphysicsequationscanbewrittenasamulti-symplecticHamiltonian system. Therefore, the research of its numerical method is of great impor-tance. Multi-symplectic structure is the intrinsic geometric property of the multisymplec-tic Hamiltonian system. It is natural to require a discretization to preserve this geometricstructure. Such a numerical algorithm is called multi-symplectic method. A great manynumerical experiments show that multi-symplectic method has significant superiority inlong time simulation compared with other method which is not multi-symplectic.The thesis is devoted to investigate multi-symplectic methods for some one-dimens-ional and two-dimensional nonlinear wave equations. The main contributions are as fol-lows:1. Multi-symplectic Fourier pseudospectral (MSFP) scheme are constructed for theCamassa-Holm equation. Moreover, multi-symplectic Fourier spectral discretization sch-eme and MSFP scheme for the KdV equation are constructed respectively. In addition, wefoundthattheIto-typecoupledKdVequationcanbewrittenasamulti-sympleticHamiltonpartial differential equation (PDE) and propose a corresponding MSFP scheme in the firsttime.2. Multi-symplectic splitting methods for the coupled nonlinear Schr(o|¨)dinger equa-tion and the two-dimensional nonlinear Schr(o|¨)dinger equation are proposed respectively.3. The MSFP scheme for two-dimensional multi-symplectic Hamiltonian PDEs isconstructed. The relevant discrete multi-symplectic conservation laws are also proved.Meanwhile, this proposed method is applied to solve the two-dimensional Zakharov-Kuznetsov equation and Kadomtsev- Petviashvili equation and the corresponding MSFPschemes are constructed.4. A great many numerical experiments are presented to show the effectiveness andsuperiority of the proposed methods in long time simulation.