The High Accurte And Conservative Numerical Schemes For Nonlinear Schr(?)dinger Equation

Firstly,In chapter one,we discuss three 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 cubic nonlinear Schr(o|¨)dinger type equations. 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 algebraic equation 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 energy 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. In chapter three,wo propose a conservative scheme that is globally linearly implicit,which means that at each discrete time level we only need to solve a set of linear 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.At last,we discuss the dissipative nonlinear Schr(o|¨)dinger equation in chapter four, propose two finite difference schemes with precision O(τ~2 + h~2).It is proved that the two schemes satisfy two conservation laws,then we give the estimate of the difference solution,prove the convergence and stability of the two schemes.At last,satisfactory numerical simulation results are obtained,It shows that there is a strong dissipative term rusulting in amplitude decay of a soliton in the time parameter.