The Convergence Of A Difference Scheme For Nonlinear Schr(?)dinger Equations

This paper discusses the high accuracy difference schemes of nonlinear Schrodinger equations,and prove the convergence of the difference schemes.Nonlinear Schrodinger e-quations are one of the most active research topics and are a basic quantum mechanics equation,it has been widely applied to various fields.There are many methods to research the numerical solution of the Schrodinger equation,whereby finite difference method is widely used by the experts at home and abroad.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 two-dimensional coupled nonlinear Schrodinger system iut+uxx+uyy+(|u|2+|v|2)u=0,(x,y,t)∈(a,6)×(c,d)×(0,T] ivt+vxx+vyy+(|u|2+|v|2)v=0,(x, y, t)∈(a, b)×(c, d)×(0, T] are numerically analyzed, a second order accuracy CN scheme is constructed.The conser-vation laws and convergence of the difference scheme are proved by energy analysis and mathematical induction.In the third chapter,the compact difference scheme with fourth-order convergence of high accuracy for one-dimensional nonlinear Schrodinger equations are discussed.We prove that the difference scheme satisfies the equation of conservation laws,and the convergence is proved in L∞norm,the order of convergence is o(τ2+h4).In the forth chapter, the linear compact difference scheme of one-dimensional non-linear Schrodinger equation is constructed. We prove that difference scheme satisfies the equation of conservation laws and have four order convergence by energy analysis.