The Average Vector Field For Several Classes Of Structure Partial Differential Equations

In mathematical physics,many partial differential equations can be written as the sym-plectic structure of Hamiltonian system or the multi-symplectic structure,such as the complex modified KdV equation,KGS equation,coupled Schrodinger Boussinesq equation,BBM equa-tion,etc.The partial differential equations have the energy conservation property.In numerical simulation,the numerical methods which can preserve the energy conservation of the partial dif-ferential equations have important meaning in simulating the behaviors of equations.In 1984,Feng Kang academician and his study team proposed the symplectic method of the Hamilton sys-tem.In 1997,based on the symplectic method,Bridges and Reich proposed the multi-symplectic method.Symplectic and multi-symplectic methods have the advantage in long time accuracy computing ability.However,these two methods only nearly preserve the energy conservation of equations.In 1999,Quispel and McLachlan constructed the average vector method which can preserve the energy conservation of the Hamiltonian system.Wang yushun constructed the en-ergy preserving method for the multi-symplectic partial differential equations based on the the average vector method.In this paper,we construct the high order energy-preserving schemes of the Hamiltonian system and the multi-symplectic global energy-preserving schemes of the multi-symplectic partial differential equation by the average vector method and the Fourier pseudo-spectral method.The new schemes are applied to simulate the partial differential equa-tion and the corresponding numerical results are investigated.In chapter 1,we construct the high order energy preserving scheme of the complex modified KdV equation by applying the fourth order average vector method in time,and Fourier pseudo-spectral method in space.And then the complex modified KdV equation is simulated by the high order energy preserving scheme.Numerical results indicate that the proposed algorithm has high accuracy,the property of structure-preservation,and numerical stability during long time computations.In chapter 2,a high order energy preserving scheme for the coupled partial differential equations is obtained by applying the high order average vector field method and the Fourier pseudo-spectral method.The high order energy preserving scheme is applied to solve the KGS equation and CSBE equations.Numerical results show that the high order scheme can simulate the solitary wave for a long time and preserve the discrete energy conservation laws well.In chapter 3,a new numerical scheme for the multi-symplectic complex modified KdV equation and the BBM equation is constructed based on the second order average vector field method and pseudo-spectral method,and the corresponding discrete global energy conservation property of the new scheme is proved.We apply the new scheme to simulate the solitary wave evolution behaviors of the complex modified KdV equation and the BBM equation with the different initial conditions and analyze the preserving energy conservation property.Numerical results show the new scheme can well simulate the solitary wave behaviors and preserve the discrete global energy conservation property of the equations.The CPU time by proposed algorithm is much less than the high order energy preserving scheme at the same time.In chapter 4,global energy-preserving schemes for the two-dimensional Zakharov-Kuznertsov(ZK)equation is obtained by applying the second order average vector field method and the Fourier pseudospectral.Numerical results show that the global energy-preserving scheme has a nice stability and well simulate the solitary wave evolution behaviors of Zakharov-Kuznetsov equations in long time and preserve the discrete energy conservation of the system.