Study Of The High Accuracy Finite Difference Method For Two Kinds Of Differential Equation

In the practical engineering problems,it is often related to this issue of differential equations,but it's difficult to get analytic solutions in solving the problems of differential equations.So the numerical solution of differential equations in the practical engineering problems plays a vital role.In many classic numerical methods,the compact finite difference method with high accuracy and conservation is widely applied,which is a research hotspot in recent years.In this paper,we study two classes of the initial-boundary value problem of nonlinear partial differential equation by using the finite difference method.At first,we give a brief overview for the research background and significance of the numerical methods of partial differential equations.And the current research status of the high accuracy finite difference method is briefly described.The research status of two classes of nonlinear partial differential equation and innovation of this paper are introduced.Secondly,a new high accuracy compact and conservative difference scheme for the nonlinear Rosenau-KDV-RLW equation is proposed.Combining with the energy analysis method,the conservative laws in energy and mass are proved.Existence and uniqueness of its difference solutions are proved.We prove that the scheme is unconditionally stable and convergence,and its numerical convergence order is O(?~2 + h~4)in the L?-norm.Numerical experiments show the reliability and validity of numerical theory.Finally,a new high accuracy compact and conservative difference scheme for the two dimensional nonlinear Schrodinger equation is proposed.Combining with the energy analysis method,the conservative laws in energy and mass are proved.Existence of the difference solutions is proved.We prove that the scheme is stable and convergence and its numerical convergence order is O(?~2 + h_x~2 + h_y~4)in the L~2-norm.Numerical experiments show the reliability and validity of numerical theory.