Structure-preserving Algorithms For The CH-? Equation

The shallow water wave equations have been widely studied because of the existence of the solitary solution.The newest shallow water wave equation--Camassa-Holm-?(CH-?)equation is a nonlinear dispersive partial differential equation.It can be written into the Hamiltonian formulation because it has symmetrical character and conservation law.The thesis aims to construct two types of structure preserving algorithm with the Hamiltonian CH-? equation.The main work is as follows.1.The multi-symplectic Hamiltonian formulation of the CH-? equation is presented for the first time.Fourier pseudospectral method is used to discrete the space direction and implicit midpoint method is applied in time discretization.We constructed the multi-symplectic Fourier pseudospectral scheme of the CH-? equation.In numerical experiments,the results show that this scheme has high accuracy and the good ability of conserving the invariants in this system.2.Regarding the character of the energy conservation of the CH-? equation,we construct an energy preserving scheme for the CH-? equation with the Hamiltonian formulation.We use the Fourier pseudospectral method to discrete the space direction and average vector field method to discrete the time direction.The numerical results show that this scheme can simulate the translations of the solitary peaked wave.It also has a good ability in the preserving of the invariants.