Modified Local Crank-Nicolson Schemes For Rosenau-Burgers Equation

It is well-known that Rosenau-Burgers equation is one of the most important model of dynamic system.It is widely used in many fields such as explosion and water wave propagation,discrete dynamics problems,wave dynamics and so on.It is difficult to solve the problem numerically because of the difficulty in processing of nonlinear term.Therefore,the study of the numerical method for Rosenau-Burgers equation is important in theoretical and practical.Firstly,the nonlinear term of the Rosenau-Burgers equation is linearized by allowing the non-linearities to lag one time step behind,The space variable of the linear Rosenau-Burgers equation carries out the central difference separation,The studied equation is transformed into ordinary differential equation.Secondly,we use the Trotter Product formula of exponential function to ap-proximate the coefficient matrix of the ordinary differential equation,Then the five diagonal sparse matrix is summed of some simple matrices according to rows and elements,and two modified lo-cal Crank-Nicolson schemes for solving the Rosenau-Burgers equations are proposed by using the Crank-Nicolson method.These explicit schemes presented In this paper are absolutely stable and has second order of accuracy for space and time.The local truncation error analysis is carried out by Taylor series expansion,and the consistency is proved.The energy inequality method is used to prove the stability and the convergence is proved.In order to verify the effectiveness of the two numerical schemes,two numerical examples are given for numerical experiments.The experimental results of the two numerical schemes in this paper are compared with the numerical results of several difference schemes in the references.It is found that the two difference schemes in this paper have obvious advantages over other numerical schemes.In terms of two difference schemes,the numerical results of the modified local Crank-Nicolson scheme by splitting the elements is better than the modified local Crank-Nicolson scheme splitting by line.These modified local Crank-Nicolson schemes by splitting in row and element are not only used to solve the Rosenau-Burgers equation,but also the numerical solutions are provided for nonlinear partial differential equations.