E Construction Of Multi-symplectic Methods And The Rror Analysis And The Dispersion Relation Analysis

In1984, Feng Kang and his study team proposed the symplectic method of the Hamil-ton system, which has a long accurately computing capability and approximately preserves the energy-preserving property of the system. Recently, based on the symplectic method, Bridges and Reich etc proposed the multi-symplectic method of PDEs. The multi-symplectic integrators, such as the multi-symplectic Preissman scheme, the multi-symplectic Runge-Kutta method, the multi-symplectic Euler box scheme, etc, which can preserve the multi-symplectic geometric structure under appropriate discretizations, have been proposed. In this paper, we mainly investigate the construction of multi-symplectic methods and the error analysis and the dispersion relation analysis.In Chapter1, the two Euler-box schemes for the saturated nonlinear Schrodinger equation are proposed. The two Euler-box schemes are combined into a new multi-symplectic scheme. The saturated nonlinear Schrodinger equation is simulated by the new multi-symplectic scheme. Numerical results show that the new multi-symplectic scheme can well simulate the solitary evolution behaviors of the saturated nonlinear Schrodinger equation, and preserves the quasi square conservation property.In Chapter2, we use multi-symplectic methods to solve three ploblems.In Section1, A new scheme for the BBM equation with the accuracy order of O(Δt2+Δx2) is proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new scheme is implemented. The solitary wave evolution behaviors are simulated by the new scheme. The new multi-symplectic scheme is compared with the Euler box scheme and the Preissman box scheme in preserving the conservation properties of the BBM equation and the computation cost. In Section2, we propose a new scheme for the generalized Kadomtsev-Petviashvili(KP) equation. The multi-symplectic conservation property of the new scheme is proved. Backward error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the KPⅠ equation and the KPⅡ equation are presented in detail. In Section3, we propose a new scheme for the Zakharov-Kuznetsov(ZK) equation. The multi-symplectic conservation property of the new scheme is proved. Backward error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the ZK equation is presented in detail.In Chapter3, based on the splitting multi-symplectic structure, a new multi-symplectic scheme is proposed and applied to the nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained and the corresponding multi-symplectic conservation property is proved. Backward error analysis shows the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and Klein-Gordon equation are simu-lated by the explicit multi-symplectic scheme. Numerical results show the new explicit multi-symplectic scheme can well simulate the solitary behaviors of the wave equation and approxi-mately preserve the relative energy error of the system.In Chapter4,we investigate the dispersion relations of the linear and nonlinear PDEs.In Section1, A new multi-symplectic Euler box scheme for the sine-Gordon equation is given and the corresponding multi-symplectic conservation property is proved. Compared with the other three classical multi-symplectic schemes, the dispersion and group velocity of the new multi-symplectic Euler box scheme for the linear PDEs are analyzed. Dispersion effects of four multi-symplectic schemes for the linear wave equation and the sine-Gordon equation to soli-tary waves are investigated. In Section2, we study the dispersive property of multi-symplectic discretizations for the nonlinear Schrodinger equations. The numerical dispersion relation and group velocity are investigated. We obtain that the dispersion property is relevant to the nu-merical solution of the nonlinear Schrodinger equations. And with the increasing of the group velocity, which is the first order derivative of the dispersion relation, the propagation velocity of the numerical solution increases.In Chapter5. vector field method (AVF) for the nonlinear Schrodinger equation which is a energy-preserving method theoretically is proposed. The nonlinear Schrodinger equation is simulated by the AVF method and the symplectic method respectively, and the energy-preserving properties for the nonlinear Schrodinger equation of two methods are compared. Numerical results show that the AVF scheme can well simulate the solitary evolution behaviors of the nonlinear Schrodinger equation, and preserves the energy-preserving property better than the symplectic scheme.