带白噪声的随机偏微分方程的Wiener混沌谱方法

Wiener expansion-spectral methods for stochastic partial differential equations with white noise

采用谱方法计算一类带白噪声的随机非线性偏微分方程.对于随机变量采用Wiener展开方法;在空间方向上,对于Dirichlet边值问题使用Legendre-Chebyshev谱方法,对于周期边值问题使用Fourier拟谱方法;在时间方向上使用二阶差分方法.针对白噪声随机项,推导了Wiener展开的计算格式.数值试验计算了随机方程解的均值、方差及高阶矩,比较原有的几个相关方法,取得较好的结果,显示了该方法的有效性.

The spectral method is applied for a class of stochastic nonlinear partial differential equations with white noise. The approximation of random is the Wiener expansion method. The spatial discretization is the Legendre-Chebyshev spectral method for the Dirichlet boundary value problem and the Fourier pseudospectral method for the periodic boundary value problem. The time discretization uses the second order finite difference method. For white noise random items, a numerical scheme is given by the Wiener expansion method. The numerical experiments compute the mean, the variance, and the higher moments of the solution. Compared with the several previous related methods, better results are obtained by the Wiener-spectral methods, which shows the effectiveness of the method.