High Order Accuracy Difference Schemes For Nonlinear Klein-Gordon Equation With Neumann Boundary Conditions

The Klein-Gordon(KG) equation, which is the relativistic version of the Schrodinger equation, plays a significant role in mathematical physics. It involves the research of some nonlinear equations. Since this problem can not be resloved in general, one can only obtain numerical solution by numerical methods. Once we applied the numerical analysis on it, it will cause certain error inevitably. Therefore, the high order accuracy methods are very necessary. In recent years, some scholars have done related researches on the high precision algorithm of the KG equation, but the correlational researches are relatively less, which use the high-order finite difference method to discrete the KG equation.This article is devoted to the study of high-order accuracy difference methods for the KG equation with Neumann boundary conditions. The paper is divided into three chapter.The first chapter introduces the practical significance and the research status of the problem which one we discuss in this paper, and states the research contents and results we get.The second chapter researches the numerical solution of the one-dimensional equation with Neumann boundary conditions. We use the boundary conditions and the equation to get the value of ux(3) and ux(5) at the boundary, so that the three point compact difference schemes and the two point compact difference schemes are respectively established at the inner points and the boundary points, which truncation errors are O(τ2+h4). Then we can analysis the schemes through mathematical induction because of the limits of g(u) and f. The stability and convergence of the difference schemes are shown by applying the energy analysis、Gronwall inequality and Schwarz inequality. After that the numerical simulations are conducted to illustrate the main results presented in this chapter.The third chapter researches the numerical solution of the two-dimensional equation with Neumann boundary conditions. We use the boundary conditions and the equation to get the value of the third order partial derivative and the fifth order partial derivative of u at the boundary, so that the high order compact difference schemes are derived and are shown by multiple multiplication. We can not analysis the schemes by mathematical induction on account of the boundness of the errors without guarantee. But it is easy to see that the truncation errors are O(τ2+h4) through the numerical simulations.