随机系数Burgers方程的广义多项式混沌-谱方法

Generalized polynomial chaos and spectral method for Burgers equation with a random coefficient

采用广义多项式混沌-谱方法求解系数随机的Burgers方程.首先,在随机方向上对随机过程和随机系数进行多项式混沌展开,采用随机Galerkin方法将随机Burgers方程化为确定性的非线性微分方程组;然后,对该方程组,在空间方向上采用Legendre-Galerkin-Chebyshev-collocation方法,对非线性项采用Chebyshev-Gauss-Lobatto点上的Legendre插值,在时间方向上采用二阶Crank-Nicolson/leapfrog格式,以保证时间方向达到二阶精度.最后,分析了该方法求解系数随机的Burgers方程的均方收敛性,并给出了数值结果.

In this paper,we solve the Burgers equation with a random coefficient by the generalized polynomial chaos expansion and spectral method.Firstly,in the random direction,the stochastic process and the random coefficient can be expanded by the generalized polynomial chaos.Then,the stochastic equation can be transformed to a system of nonlinear equations by the stochastic Legendre-Galerkin method.The Legendre-Galerkin-Chebyshev-collocation scheme is used in the direction of space,that is,the nonlinear term is interpolated through the Chebyshev-Gauss-Lobatto points.In the direction of time,to ensure the second order accuracy,the second-order Crank-Nicolson/leapfrog scheme is used.Finally,the numerical results are given,and we analyse the mean square convergence of this method.