Some Numerical Research On Nonlinear Schr(?)dinger Equations

Difference schemes are given, based on the idea of finite difference, to solve two kinds of nonlinearSchr dinger equation.In chapter1, we recall why the problems in this paper are studied and the existing results for them,then we describe how this paper is organized and the main work of this paper.In chapter2, firstly, three alternating direction implicit (ADI) schemes with second-order accuracyin space direction are proposed for two-dimensional nonlinear Schr dinger equation. The first schemewe give is of two time levels, which needs iterating to solve and is time-consuming. Then a three-time-level ADI scheme is presented in avoid of iteration, another linearized ADI scheme which needn’titerate to solve is given next. With compact difference operator, we get another three high-ordercompact ADI (HOC-ADI) schemes with four-order accuracy in space direction. The unconditionalstablility character is verified by linear of stable analysis method. All the schemes can be used to solvelinear Schr dinger equation condition. The convergence rate and unconditional stablility character aredemonstrated numerically.In chapter3, we study the derivative nonlinear Schr dinger equation, and give a relaxationdifference scheme. The scheme uses the basic idea of the over-relaxation iteration method, for thecalculation of each step, the residual value of error is made up to the latter step, meanwhile, we add aintermediate variable, so without lowering the accuracy of the calculation, we save muchcomputational time. The scheme keeps conservation laws. We give another Crank-Nicolson (C-N)scheme and linear C-N scheme to compare with the relaxation scheme. The extensive numericalresults verify the accuracy and efficiency of the relaxation scheme.