With Five Nonlinear Schr (?) Dinger Equation Finite Difference Method

This article raises a kind of conservative diference scheme of the nonlinearSchro¨dinger equation with involving quintic term. Then proves its stability andconvergence. At the same time, we analyse the precision of this scheme.In the first part, which is introduction, the historical development and someresearching achievements of the nonlinear Sch¨odinger equation with involving quinticterm are simply introduced. We give the relationship between the discrete energyand charge, then prove it.In the second chapter, we give two steps diference scheme of the nonlinearSchro¨dinger equation with involving quintic term and estimate its solution. First-ly, we give the diference scheme, and estimate its diference solution. Secondly, weprove the convergence and stability of the diference scheme by using energy method,the precision of this scheme is O(h2+τ2). Finally, we summarize the above cal-culation method. We compared this diference scheme with pre-existing schemeby numerical experiments§the results show that this scheme not only conservethe discrete energy and charge of systems but also have convergence and stability.Moreover, this scheme improves the computational accuracy.In the third chapter, we give three steps diference scheme to the nonlinearSchro¨dinger equation with involving quintic term and estimate its solution. First ofall, We propose a conservative scheme that is globally linearly algebraic equations.Then we estimate its diference solution. Finally, we prove the convergence andstability of the diference scheme by using the energy method, the precision of thisscheme is O(h2+τ2). This scheme improves the computational accuracy and thespeed of the computation.