Conservative Orthogonal Spline Collocation Schemes For Solving Coupled Klein-Gordon-Schr(?)dinger Equations

In recent years, orthogonal spline collocation methods have been studied extensively for the numerical solutions of various problems of differential equations. However, it's still very little to apply these methods to those coupled nonlinear differential equations. Therefore, this thesis is trying to use these methods to solve a problem of coupled Klein-Gordon-Schr?dinger equations with initial and boundary conditions.A time-continuous and a class of time-discrete orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schr?dinger equations with initial and boundary conditions are considered herein. The continuous-time scheme is made by using orthogonal spline collocation methods for spatial discretization, and the discrete-time schemes are constructed by applying finite difference methods to the time-continuous scheme for temporal discretization. These schemes are proved to have the conservative laws of discrete energy which preserve the conservations of the original problem. They are both showed to possess optimal order in L2 -norm for space, and second order accuracy in maximum norm for time for discrete-time schemes. At last, several numerical experiments are carried to test the performance of the discrete-time schemes. It's verified numerically that those schemes (i) exhibit second order accuracy for time and fourth order for space, and (ii) have two conservative laws, and (iii) preserve the soliton shapes well during the propagations of the soliton waves, which accord with the theoretical analysis.