Theoretical And Numerical Methods Of Kuramoto-Sivashinsky Equation

In this paper,we investigate the Kuramoto-Sivashinsky equation(KSE)by mean of numerical methods and analysis.The main purpose is to construct,compare,and analyze several numerical schemes for the KSE.The content includes:Firstly,we recall some basic facts about the KSE in bounded domains,and give the motivation of the research carried out in this paper.Secondly,we review the stability result of the equation,especially the influence of the anti-diffusion coefficient on the stability.The main body of the paper is concerned with construction and analysis of several numerical schemes for KSE.The proposed methods combine a number of finite difference schemes in time and spectral approximation in space.The convergence analysis is carried out to derive error estimates.In order to be more robust,a C1-spectral element method is also proposed.Finally,we provide a series of numerical examples to verify the convergence rate of the methods,and study the stability of the solution with respect to the anti-diffusion parameter.We emphasize that in our calculations,a fast solver based on the sparse large matrices solution algorithm is used.The main idea of the algorithm is to split the computation into the interior nodes and interface nodes so that the overall system is reduced to a set of smaller systems.