Several Numerical Schemes Of Nonlinear Schrodinger Equation With Quintic Term

In this article,several numerical schemes are proposed to solve the given Schr(?)dinger equation.The main idea is time splitting method.Two schemes are established by time splitting combining with compact difference.The rest scheme is time splitting combining with pseudo-spectral method.In chapter 1,some preparations are introduced so that we can easily understand the content behind this chapter,including the background of the given question and introduce of the splitting method.In chapter 2,time splitting method combined with compact difference is used to create two numerical formats.At first,we spilt the given problem into two subproblems so that one subproblem need to be solved with numerical method and the other can be solved exactly.Depending on the different splitting method,the two schemes are of different accuracy in time.The order of the scheme with Lie-type is one in time and four in space and the order of the scheme with Strang-type is two in time and four in space.This convergence is proved by theory and so is mass conservation.In the end of this chapter,we do some numerical tests.And the result is consistent with the previous conclude.In chapter 3,we still use time splitting method.The difference is the order in space.Instead of four order,we proposed a scheme of six order in space.The convergence is confirmed in theory and numerical experiment,and so is the stability of the numerical solution.In chapter 4,we discrete the second derivative in pseudo-spectral method.By this and time splitting method,we proposed a new scheme which is stable.We show the answer of the scheme is almost same as the answer of the given problem.In chapter 5,we make a summary of the previous chapters.By compare the whole schemes,we can choose the most suitable scheme at the given conditions.