Finite Difference Method Of Generalized Rosenau Equation

It is well known that conservative difference scheme is better than nonconservation difference scheme. Zhang fei pointed out that non-conservation diffeence scheme can appear nonlinear blasting phenomena easily. Li and Vu-quoc pointed out "in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation."The generalized nonlinear Sehrodinger equation, regularized long wave equation, Sine-Gordon equation, Klein-Gordon equation and Zakharov equation have been solved by some conservative difference scheme, recently, and richer numerical results have been obtained.Difference scheme is constructed by stimulating energy conservation law of set solution, such difference scheme is called conservative scheme.In this paper, some new conservative difference scheme has been prospered to solve generalized Rosenau equation. Parameter p is contained in nonlinear item of generalized Rosenau equation, which is famous Rosenau equation when p=1. The generalized Rosenau equation has more universal than Rosenau equation, however, there is no report about exact solutions of generalized Rosenau equation, so its numerical solution is very important.Several conservative difference schemes of initial boundary value problem of generalized Rosenau equation have been proposed in this paper, and two two-level nonlinear finite difference schemes and a three-level linear finite difference scheme are given by Taylor series expansion, respectively, and the existence of the solution of difference scheme is proved by Brouwer fixed point theorem, the convergence and the stability of these difference schemes are proved by discrete energy analysis.