The High Accurte And Conservative Numerical Schemes For Nonlinear Schr(?)ding Equation Involving Quintic Term

Firstly, In chapter one, We discuss two main numerical methods for the nonlinear Schr(o|¨)dinger equation, review the previous results and present the primary lemmas.Secondly,We present two high accurate and conservative difference schemes for nonlinear Schr(o|¨)dinger equation involving quintic term.In chapter two, We present a globally nonlinear implicit difference scheme,i.e., a nonlinear iterative algorithma has to be used to solve the system of the nonlinear at each discrete time step. The scheme can conserve the energy and charge of systems,and its convergence and stability are proved by using the enery method,the precision of this scheme is O (τ~2 + h~4). By means of numerical computing, We get the conclusion that the scheme in this chapter has higher precision than the other schemes.Finally, In chapter three, We propose a conservative scheme that is globally linearly algebraic equations. As a consequence our scheme is faster and simpler than the nonlinear implicit difference schemes. The conservation of the energy and charge can also be proved.We show rigorously that our scheme is stable and convergent and that it will not yield"blow-up", the precision of this scheme is also O (τ~2 + h~4).The numerical tests show that the new scheme is faster and more accurate than the other conservative schemes.