We consider the numerical approximation of different ordinary differential equations (ODEs) and partial differential equations (PDEs) with periodic boundary conditions involving a one-dimensional random parameter, comparing the intrusive and non-intrusive polynomial chaos expansion (PCE) method. We demonstrate how to modify two schemes for intrusive PCE (iPCE) which are highly efficient in solving nonlinear reaction-diffusion equations: A second-order exponential time differencing scheme (ETD-RDP-IF) as well as a spectral exponential time differencing fourth-order Runge-Kutta scheme (ETDRK4). In numerical experiments, we show that these schemes show superior accuracy to simpler schemes such as the EE scheme for a range of model equations and we investigate whether they are competitive with non-intrusive PCE (niPCE) methods. We observe that the iPCE schemes are competitive with niPCE for some model equations, but that iPCE breaks down for complex pattern formation models such as the Gray-Scott system.

翻译：本文考虑求解包含一维随机参数的周期边界条件下的常微分方程和偏微分方程的数值逼近问题，对比了侵入式与非侵入式多项式混沌展开方法。我们展示了如何将两种高效求解非线性反应扩散方程的格式改造为适用于侵入式多项式混沌方法：即二阶指数时间差分格式（ETD-RDP-IF）和谱指数时间差分四阶龙格-库塔格式（ETDRK4）。数值实验表明，对于一系列模型方程，这些格式比EE格式等简单方法具有更优的精度，同时我们探究了它们与非侵入式多项式混沌方法的竞争性。研究发现，对于部分模型方程，侵入式多项式混沌格式与非侵入式方法相当，但对于格雷-斯科特系统等复杂斑图形成模型，侵入式方法会失效。