Study On High Order Finite Difference Methods For Schr?dinger Equations

In scientific calculation,finite difference method is commonly used to solve vari-ous partial differential equations.It is one of the widely used numerical methods.In this paper,we study several high accuracy numerical methods for solving the non-linear Schr?dinger equations based on the existing results.Nonlinear Schr?dinger equation plays a powerful role in physical applications,especially in fluid mechan-ics,nonlinear optics,quantum mechanics,etc.However,it is difficult to obtain the exact solution of multi-dimensional nonlinear Schr?dinger equations and fractional Schr?dinger equations.Therefore,it is an important task to establish some con-servative finite difference schemes to solve multi-dimensional nonlinear Schr?dinger equations and fractional Schr?dinger equations.At present,due to the advantages of high accuracy and high efficiency,high-order compact finite difference schemes have attracted more and more attention of researchers at home and abroad.In this thesis,for multi-dimensional nonlinear Schr?dinger equation,we con-struct some conservative high order compact difference schemes and analyze the conservation and stability of the difference scheme.For fractional Schr?dinger equa-tion,we also construct several high-order compact difference schemes,and the nu-merical analysis of the schemes is carried out.The specific research content of the whole thesis includes the following six parts:In Chapter 1,we introduce the research background of the nonlinear Schr?dinger equation and the fractional Schr?dinger equation and the research status at home and abroad.We also introduce the development of the high order compact differ-ence scheme and describe the main work of this thesis.Finally,some of the basic kno,wledge to be used in subsequent chapters is reviewed.In Chapter 2,we give the fourth-order compact split-step difference schemes for the two-dimensional and three-dimensional nonlinear Schr?dinger equations.In this chapter,in order to overcome the difficulty caused by nonlinear problems,we use the operator splitting technique to solve the original equations.Splitting into a linear subproblem and a nonlinear subproblem.A conservative fourth-order com-pact difference scheme is established for linear subproblems and the stability and conservation of the scheme are discussed.The nonlinear subproblem can be solved accurately.Numerical examples have verified the accuracy and efficiency of the scheme we construct.In Chapter 3,We study the numerical solut,ion of the three-dimensional non-linear Schr?dinger equation.In order to resolve the difficulties of solving the multi-dimensional solution,we use the split-step method to construct the fourth-order and sixth-order compact alternating direction implicit(ADI)difference schemes respec-tively,and prove that the two schemes are uncondit,ionally Stability.The discrete conservation properties,accuracy and stability of the two schemes are verified by numerical experiments.In Chapter 4,we propose a fourth-order compact difference scheme and a compact ADI difference scheme for the one-dimensional and two-dimensional time-order Schr?dinger equations.The time fractional Schr?dinger equation contains ?(??(0,1))order-Caputo time fractional derivative.In this chapter,the L1 numer-ical formula and the L1-2 numerical formula are used to approximate the Caputo time fractional deriva.tive,and the apply the fourth-order compact difference for-mula for the space direction derivative.The stability of the scheme is proved by Fourier analysis and mathematical induction method,and the theoretical analysis results are tested by some numerical examples.In Chapter 5,On the basis of previous work,we study a conservative split-step Crank-Nicolson difference scheme for solving the fractional Schr?dinger equation.In addition,we give the stability and convergence analysis of the algorithm.Fi-nally,numerical examples are given to verify the efficiency of the algorithm and the accuracy of the theory.In Chapter 6,we give a summary of our work and a prospect for future work.