求解Kuramoto-Sivashinsky方程的有限体积二次元方法

Quadratic Finite Volume Element Method for the Solution of the Kuramoto-Sivashinsky Equation

Kuramoto-Sivashinsky方程是一个非线性四阶偏微分方程,常被用于反应扩散系统的动力过程建模。在文章中,首先引入一个新变量,将方程转化为一个低阶的方程组,然后采用有限体积二次元方法对其空间变量进行离散近似,时间积分采用显式2阶Runge-Kutta格式,在数值实验中采用所提出的方法分别对激波解以及混沌现象进行数值模拟。结果表明提出的有限体积元方法能够对以上各种现象进行有效模拟。

The Kuramoto-Sivashinsky equation belongs to nonlinear four-order partial differential equations(PDEs),which is commonly used for the modelling of dynamics in reaction-diffusion systems.In this paper,a new variable is introduced and the K-S equation can be converted into low-order equations.The finite volume quadratic element method is used to approximate the space derivatives of the equations.As for the time integration,explicit two-order Runge-Kutta scheme was utilized,Numerical experiments,including two travelling shock waves,a chaotic would be simulated by the proposed method respectively.The results demonstrate that our method could simulate all the above experiments successfully.