A High-order Compact Difference Scheme For Nonlinear Schr?dinger Equation

In this thesis,we mainly study the compact finite difference scheme and numerical calculation for nonlinear Schr(?)dinger equation(system)on an unbounded domain.Schr(?)dinger equation reveals the law of the state of micro particles changing with time.Schr(?)dinger equation not only has a clear quantum mechanical significance,but also has been widely used in many scientific fields,such as the propagation of optical fiber,solid state physics,life science,as well as Bose-Einstein condensation,superconductivity and deep water wave and other important fields.Nonlinear Schr(?)dinger equation is a nonlinear equation with a soliton solution.These years,many researchers have been devoted to obtain the numerical solution of nonlinear Schr(?)dinger equation on an unbounded domain.To our knowledge,the difficulties for numerical solutions of this kind of problem defined on unbounded domains are the unboundedness of the physical domain and the nonlinearity.Currently,artificial boundary method is one of the effective methods to overcome this challenge.By setting appropriate artificial boundaries,the original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain with the artificial boundary conditions.Then,the reduced problem on the bounded computational domain is solved by an efficient semi-implicit compact finite difference scheme,which is a fourth-order scheme with respect to spatial variable.The scheme efficiently avoids the time-consuming iteration procedure necessary for the nonlinear method and improves the rate of convergence.We develop a semi-implicit compact finite difference scheme that conserves the mass and the energy in the discrete level for the Zakharov-Rubenchik equations,and some discrete conservation laws of the scheme are proved.Finally,some numerical examples are given to demonstrate the accuracy,conservation and effectiveness of the proposed scheme,and the comparison of difference schemes in the errors and time-consuming is carried out to illustrate the efficiency of semiimplicit compact finite difference scheme.