High Order Exponential Compact Difference Method For Convection-Diffusion-Reaction Equation With Complex Coefficients

Convection-diffusion-reaction equation is a class of important differential equation,which can describe many problems in the field of natural and engineering.Most of the traditional difference method and the finite element method and spectral method are mainly aimed at real coefficient or real variable convection diffusion equation,then in this paper we mainly discuss the complex(variable)coefficient convection-diffusion-reaction equations.Firstly,for one-dimensional complex variable coefficient convection-diffusion-reaction equations,we construct the second order exponential difference scheme,based on the format we build the fourth order accuracy compact difference scheme with exponential by applying the method of residual correction,for matching with interior point difference scheme,we construct a four order accuracy of boundary condition and a numerical example is given to verify the high precision and effective of the method.Secondly,the four order compact difference scheme for the two-dimensional complex(variable)coefficient convection-diffusion-reaction equation is established by using the dimension reduction method.Finally,as an application of the method,we discuss the nonlinear Schr?dinger equation by using the method of time splitting high accuracy difference.At first,the original equation with nonlinear boundary is transformed into the N eumann boundary conditions,and calculate by the time splitting method,then the nonlinear part can be calculated precisely and the linear part of the space can be dispersed by using the provided four order compact difference scheme.Numerical experiments are given to prove the method with fourth order accuracy is simple,effective and economical.