Application Of Average Vector Field Method In Coupled Partial Differential Equations

Nonlinear phenomena is a common dynamic behavior in applied mathematics and physics.It can be described by many coupled partial differential equations,such as KdV-mKdV equation,KdV-ZK equation,KdV-Burger equation and coupled Schrodinger-KdV equation.The system which described by these coupled PDEs have energy conservation properties.In 1984,Symplectic geometry symplectic geometry algorithm is proposed by Feng Kang for the first time.Later,Bridges and Reich proposed the multi-symplectic algorithm.In order to keep Hamilton's conservation of energy accurately,Quispel and McLachlan proposed the average vector field method in 1999 and is widely used in various PDEs.Based on the correction of the average vector field method,Quispel and McLachlan proposed a high-order energy-preserving energy format with third-order and fourth-order accuracy in time.In this paper,we use the average vector field method and Fourier pseudo-spectral method to construct the high-order guaranteed energy format of coupled partial differential equations.The new format of the equation is numerically simulated and its numerical results are analyzed.In chapter 1,in section 1,the high order energy preserving scheme of 3-coupled Schrodinger equations is obtained by the fourth-order average vector field method.The new scheme is applied to simulate the solitary wave behaviors of 3-coupled Schrodinger equations with different parameters.The preserving energy conservation property of the scheme is also analyzed.In section 2,The high order energy preserving scheme of the coupled Schrodinger-KdV equation is obtained by the fourth-order average vector field method and Fourier pseudo spectrum method.And simulate the behavior of solitary wave with the new format.In chapter 2,based on the fourth order average vector field method and the Boole discrete line integral theory,we propose the high order Boole discrete line integral method of the Hamiltonian system.The high order Boole discrete line integral method is applied to the energy conservation coupled Schrodinger-KdV equation.A new high order scheme of the energy conservation this equation is obtained.In chapter 3,based on energy conservation theory in the study of global multi-symplectic,the average vector field method and Fourier pseudo-spectral method is applied to numerically discrete the multi-symplectic global energy preserving scheme of 3-coupled Schrodinger equations is proposed.The numerical solution of of 3-coupled Schrodinger equations is obtained.