中文摘要：

本文针对对流一扩散随机过程在随机输入（即随机输运和源项），作用下进行数值仿真。我们先将对流扩散随机微分方程中的随机函数采用有限项截断的多项式浑沌展开（Polynomial Chaos Expansion）展开，再由Galerkin映射法得到求解浑沌展开系数的确定性方程组。这是一个在物理空间包含多尺度解的大方程组。为此我们发展了多重网格求解器，在不同尺度网格叠代求解。给出了：1）有精确解的算例，以检验求解器的收敛性和精度；2）随机流场中的浓度对流扩散过程的数值模拟。

英文摘要：

We apply stochastic non-statistical approach to numerically simulate the convectiondiffusion processes under uncertain inputs, i.e. random flow （transport） velocity or/and source （forcing） term. We first represent the random functions involving in the stochastic partial differential equations in terms of the truncated polynomial chaos expansion, then perform Galerkin projection to obtain a coupled deterministic system of equations for the coefficients of the expansion. Due to the spectral representation, the size of the system is much larger than the deterministic counterpart, and moreover, it is multi-scale in nature. In this work, we develop a multi-grid solver to iteratively solve the system on different levels of mesh. We present two numerical examples： the first is a test problem with exact solution examining the accuracy and convergence of the solver, the second simulates the convection and diffusion process in the concentration field under random flow （transport） velocity.