求解Kuramoto-Sivashinsky方程的平移基无单元Galerkin方法

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

Kuramoto-Sivashinsky方程是一种可以描述复杂混沌现象的高阶非线性演化方程.方程中高阶导数项的存在,使得传统无单元Galerkin方法采用高次多项式基函数构造形函数时,形函数违背了一致性条件.因此,本文提出了一种采用平移多项式基函数的无单元Galerkin方法.与传统无单元Galerkin方法相比,该方法在方程离散时依然采用Galerkin进行离散,但形函数的构造采用了基于平移多项式基函数的移动最小二乘近似.通过对具有行波解和混沌现象的Kuramoto-Sivashinsky方程的数值模拟,验证了本文方法的有效性.

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.