【作者】 李宁；

【导师】 冯新龙；

【作者基本信息】 新疆大学 ， 数学， 2015， 硕士

【摘要】 本文主要采用广义多项式混沌方法求解带有随机输入的热传导方程和带有随机输入的Allen-Cahn方程,通过多项式混沌方法对随机参数空间的处理,一个带有随机输入的偏微分方程就转化为了一族耦合的确定性方程组.本文在时间上采用混合的一阶和二阶显隐格式,在空间上采用Fourier谱方法对这族方程组进行求解.对于带有随机输入的热传导方程,均值误差和方差误差均可以达到时间一阶精度和二阶精度,并且随着多项式混沌最高阶数的提高,两种误差均呈递减的趋势.对于带有随机输入的Allen-Cahn方程,我们对非线性项采用显式格式,均值误差可以达到时间一阶精度和二阶精度,而方差误差的精度却是递减的趋势,通过与蒙特卡洛方法数值结果的对比,多项式混沌方法依然保持了高效性.更多还原

【Abstract】 We discuss in this paper solvers for stochastic heat conduction equation and stochastic Allen-Cahn equation in random media, and the stochastic partial differential equation can turn into a set of coupled deterministic partial di?erential equations by using the polynomial chaos method. Here, we employ mixed ?rst order and second order explicit/implicit scheme in temporal and adopt fourier spectral method in spatial to solve the set of deterministic partial di?erential equations. For the stochastic heat conduction equation, the error of mean and variance will reach ?rst order accuracy and second order accuracy, respectively,and the two kinds of error getting smaller with the order of polynomial chaos increases. For the stochastic Allen-Cahn equation, we employ explicit scheme for the nonlinear term and the error of mean have ?rst order accuracy and second order accuracy, but the error of variance is not well. Moreover, numerical comparisons with the Monte Carlo method are reported, which demonstrates the e?ectivity and feasibility of the polynomial chaos method.更多还原