Study On Numerical Methods For The Regularized Long-Wave Equation And The Rosenau-Burgers Equation

In this paper, a new finite difference method for initial-boundary value problem of Regularized Long-Wave equation is considered. Two energy conservative finite difference schemes, a two-level linear scheme and a three-level implicit scheme, are proposed. And the prior estimate of the solution is obtained. Existence and uniqueness of numerical solution are derived. Convergence and stablity of the finite difference schemes are proved by Discrete Sobolev's inequality and Discrete Gronwall's inequality. Numerical experiment results show that the theory is accurate and the schemes are more reliable and efficient than others. By using two difference schemes for solving Regularized Long-Wave equation at different time levels, we have good conservative results. When choosing appropriate weight coefficient, the errors of solution will be greatly improved.We also make a research on the numerical method of Rosenau-Burgers equation. Rosenau equation including the viscous term -uxx is the Rosenau-Burgers equation. Firstly, we introduce and summarize the study on numerical methods for the Rosenau equation combined with previous work. Secondly, a finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-Burgers equation is proposed. And the prior estimate of the solutions was obtained. It is proved that the finite difference scheme is convergent and stable by Discrete Sobolev's inequality and Discrete Gronwall's inequality. Thirdly, numerical results demonstrate that the scheme is efficient and accurate. Finally, we introduce our research on study of numerical methods for the generalized Rosenau-Burgers equation. It can make the research topic of this article more comprehensive and more perfect. In addition, our research can be applied in the traffic flow model. We will give a brief introduction as follows.