Combined High-Order Compact Schemes For Multiple Spatial Derivatives Differential Equations

For differential equations with multiple spatial derivatives,there are some shortcomings in discretizing the classical high-order compact method,which will reduce the computational efficiency of the scheme.Based on this,this thesis designs a new combined high-order compact scheme,which successfully overcomes this shortcoming.The combined high-order compact difference method of the first-order derivative and the second-order derivative is studied,and the combined high-order compact difference scheme of the first-order derivative and the third-order derivative is constructed at the same time.Use these schemes to discretize the KdV equation and the nonlinear Schr?dinger equation in space.For the nonlinear Schr?dinger equation,it is divided into two sub-problems,and the nonlinear sub-problem is solved accurately;for the linear subproblem,CHOC-CNLS was constructed by using the combination of first and second derivatives in space and CHOC-CNLS in time using Crank-Nicholson method.The CHOC I and CHOC II schemes are constructed by using the combined high-order compact difference method of the first and third derivatives in space and the first and second order discretization methods in time,respectively.Finally,some numerical experiments are given to verify and discuss the scheme.The arrangement of the article is as follows:In Chapter 1,the research background and significance of finite difference methods are introduced,and the traditional high-precision difference schemes,high-order compact difference schemes and combined high-order compact difference schemes are analyzed and compared.At the same time,the research background and significance of the KdV equation and the nonlinear Schr?dinger equation are explained.In Chapter 2,the basic idea of combined high order compact difference method and the combined high order compact difference method of first and second derivatives are described,and the corresponding difference operator matrix is given.Then the nonlinear Schr?dinger equation is split into two sub-problems,and the nonlinear sub-problems are solved accurately;for the linear sub-problems,the combined highorder compact difference method is used in space,and the Crank-Nicholson method in time is used to construct CHOC-CNLS scheme,and analyze the truncation error and conservation of the scheme.Finally,the numerical solution of the constructed scheme is simulated through numerical experiments,and its accuracy and error are calculated,and a chart analysis is given.In Chapter 3,the combined high order compact scheme of the first derivative and the third derivative is constructed,and the corresponding difference operator matrix is given.the CHOC I and CHOC II schemes are discretely constructed by using the combined high-order compact difference method in space and the firstorder and second-order difference methods in time respectively for the KdV equation,and the truncation errors of the two schemes are analyzed.Finally,the numerical solution of the constructed scheme is simulated through numerical experiments,and its accuracy and error are calculated,and a chart analysis is given.In Chapter 4,summarize the work done in the thesis and look forward to the future work.