Efficient Numerical Computations of 
Stochastic Partial Differential Equations

 

Abstract

Stochastic PDE's with solutions depending on multiple scales play fundamental and important roles in many problems such as composite materials, flows and transports in porous media, and turbulence. Numerical simulations become an important strategy in gaining understandings to the phenomena and exploring their applications. However, direct numerical simulations are often very difficult due to the problem's multiscale nature and randomness. In this talk, I will report two different approaches that we have been explored recently. One is using Wiener Chaos expansions which separate randomness from the problems, to convert random problems into deterministic ones. Therefore, by solving these deterministic equations, all statistical properties, such as mean and variance of the original problems can be fully recovered. We have demonstrated that this approach can be applied to a wide range of problems.  The other approach is to use a dynamic nonlinear transformation and to characterize the probability density functions (PDF) of the transformed random variable by using Fokker-Planck equations. This enables us to compute the desired statistical properties efficiently and accurately using quadratures. Both approaches need not involve any randomness in the computations, thus avoiding, e.g., random number generating. Therefore we can use well developed deterministic techniques to solve nonlinear stochastic differential equations. In many applications, they can drastically reduce the computation load and provide reliable control over the computational errors.