High Accuracy Finite Difference Schemes Of Nonlinear Schr(?)dinger Equations

In this paper,we discuss the high accuracy difference schemes of nonlinear Schrodin-ger equations,three different kinds schemes of two classes Schrodinger equations are con-structed,and prove the convergence of the difference schemes.The Schrodinger equation is one of the most important equations in mathematics and physics,which is used in many fields.The paper discusses different difference schemes for nonlinear Schrodinger equations,we use energy method to prove convergence and conservation laws of difference schemes. This report consists four chapters.The first chapter is an introduction.The research background and current situation of the problem are briefly introduced,some denotations and lemmas and research results are described.In the second chapter,a coupled nonlinear Schrodinger equations are numerically analyzed. A three-level of four order accuracy CN scheme is constructed. The conservative properties and convergence in maximun norm with order O(τ2+h4) are proved by energy method.In the third chapter,we discuss one-dimensional nonlinear Schrodinger equations Constructing two-level CN scheme. Using energy method to prove the two-level scheme satisfies conservative properties, and has high order convergence in L∞norm.In the forth chapter, the three-level linear compact difference scheme of one dimensio-nal nonlinear Schrodinger equations are constructed.We prove that difference scheme satisfies the conservation laws and has four order convergence in L∞norm.