Research For The Numerical Methods For Two Types Of Nonlinear Long Wave Equations

The Rosenau equation is a typical nonlinear long-wave equation used to describe the wave-wave and wave-wall interactions.The regular long wave(RLW)equation is another form of the nonlinear long wave equation.The Rosenau-RLW equation is exactly the generalized form of Rosenau equation and RLW equation.As the further consideration for nonlinear long wave,we need to add the viscosity termsxxx(10)u andxxxxx-u to the Rosenau-RLW equation to obtain Rosenau-Kd V-RLW equation and Rosenau-Kawahara-RLW equation.However,there are few analytical solutions for these two types of long wave equations.The numerical solution is very meaningful.Based on the study of special cases in the existing literature,this paper develops numerical methods for general situations.Firstly,a two-level nonlinear difference scheme with second-order theoretical accuracy is proposed for the initial-boundary value problem of the generalized Rosenau-Kd V-RLW equations.The scheme simulate two conserved properties of the problem,it is obtained that the existence and uniqueness of those finite differences solution.The discrete functional analysis method is used to analyze the convergence and unconditional stability of the difference scheme.The results attained for numerical experiments.In addition,based on the numerical solution of the general Rosenau-Kd V-RLW equation and the improvement of the numerical discretization method at the same time,we propose two nonlinear schemes with second-order theoretical precisions for the initial boundary value problem of the generalized Rosenau-Kawahara-RLW equation.They are obtained that the existence and uniqueness of those finite differences solution.The discrete functional analysis method is used to analyze the convergence and unconditional stability of the difference scheme.