The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation

Abstract

The Kuramoto-Sivashinsky equation is a kind of high-order nonlinear evolution equation which can describe complicated chaotic nature. Due to the existence of high-order derivatives in the equation, the shape functions violate the consistency conditions when using traditional element-free Galerkin method which adopts high-order polynomial basis functions to construct the shape functions. In order to solve the problems encountered in the traditional element-free Galerkin method, a kind of element-free Galerkin method adopting the shifted polynomial basis functions is presented in this paper. Compared with the traditional element-free Galerkin method, the Galerkin principle is still used to discrete the equation in this method, but the shape functions are constructed by moving least squares based on the shifted polynomial basis functions. Numerical results for the Kuramoto-Sivashinsky equation having traveling wave solution and chaotic nature prove the validity of the presented method.