Study On Numerical Methods For Some Nonlinear Equations

Finite difference method,also known as grid method,is one of the commonly used methods for solving nonlinear equations.The main process is to construct a reasonable difference scheme,and the approximate solution of the difference scheme preserves some main properties of the original problem.It is necessary to point out that the high approximation accuracy difference scheme does not necessarily get a very a good approximate solution,because a reasonable difference scheme must also retain the physical properties of the original problem.Therefore,to construct a reasonable numerical schemes based on maintaining the original physical laws of the nonlinear equation is of great significance.The rest of this paper is organized as follows:In Chapter 1,we introduce the background of the research objects and the main work of this paper.In Chapter 2,numerical solutions for the Rosenau-KdV equation coupling with the Rosenau-RLW equation are considered and a new C-N pseudo-compact conservative numerical scheme,which preserves the original conservative properties is designed.The proposed scheme is based on a finite difference method.The existence of the difference solutions has been shown by the Brouwer fixed point theorem.Unconditional stability,second-order convergence,and a prior error estimate of the scheme are proved by the discrete energy method.Numerical examples have been given to verify the theoretical results.In Chapter 3,we develop a high-order conservative numerical scheme to solve the initial-boundary problem of GRLW equation.The proposed scheme is threelevel and linear-implicit based on a finite difference method.A detailed numerical analysis of the scheme is presented including a convergence analysis result.Some numerical examples are provided to show the present scheme is efficient,reliable,and of high accuracy.In Chapter 4,we design a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation.The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical“energy”of the system and finite difference method.The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time.The unique solvability of numerical solutions is shown.A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory.Numerical results are given to show the validity and the accuracy of the proposed method.In Chapter 5,a high-accuracy conservative difference scheme is presented to solve the initial-boundary value problem of the Zakharov equations,which preserves the original conservative properties.The proposed scheme is based on finite difference method.The scheme is second-order accuracy in time and fourthorder accuracy in space.A detailed numerical analysis of the scheme is presented including a convergence analysis result.Numerical examples are given to confirm the proposed scheme is efficient,reliable and of high accuracy.In Chapter 6,we summarize the main work done in this paper,put forward the follow-up research conception and target in the further research.