High Order Difference Schemes For A Class Of Linear Hyperbolic Equations With Neumann Boundary Conditions

This paper is devoted to the finite difference method for solving a class of linear hyperbolic equations with Neumann boundary conditions.The article is divided into three parts.The first part is an introduction,which mainly introduces the practical significance and the research status of the problem,and states the contents and results of research in this paper.The second part includes chapter 2 and chapter 3.In the chapter 2,a high order difference scheme is established for one-dimensional linear hyperbolic equation with Neu-mann boundary conditions.The value of ux(3)and ux(5)at the boundary can be got by using the boundary conditions and the equation,then the three points and two points compact difference schemes are respectively established at the inner points and the boundary points by using the finite difference method.Then the priori estimate of the difference scheme is given by using the energy estimate method,Gronwall inequality and Schwarz inequality.Finally,the stability and convergence of the difference scheme are shown.The convergence order in maximum norm of the difference scheme is O(?2 + h4).In chapter 3,a high order difference scheme is established for two-dimensional linear hyperbolic equation with Neu-mann boundary conditions using the same discrete method.To obtain the convergence and stability of the numerical solution in maximum norm,a new norm is introduced to limit maximum norm.Then two priori estimates of the difference scheme are shown.When the convergence of the difference scheme is shown,the differential mean value theorem is used to deal with the right item,so that the convergence orders of the right item in H1 norm and L2 norm are the same.The convergence order of the difference scheme in maximum norm is O(?2 + h4).Four numerical examples are presented in the third part.The numerical examples illustrate the convergence of the high order difference schemes and the convergence orders are 0(?2 + h4)in one dimension and two dimension.