Adaptive Stroud Stochastic Collocation Method for Flow in Random Porous Media via Karhunen-Loeve Expansion

In this paper we develop a Stochastic Collocation Method (SCM) for flow in randomly heterogeneous porous media. At first, the Karhunen-Loeve expansion is taken to decompose the log transformed hydraulic conductivity field, which leads to a stochastic PDE that only depends on a finite number of i.i.d. Gaussian random vari- ables. Based on the eigenvalue decay property and a rough error estimate of Stroud cubature in SCM, we propose to subdivide the leading dimensions in the integration space for random variables to increase the accuracy. We refer to this approach as adap- tive Stroud SCM. One- and two-dimensional steady-state single phase flow examples are simulated with the new method, and comparisons are made with other stochastic methods, namely, the Monte Carlo method, the tensor product SCM, and the quasi- Monte Carlo SCM. The results indicate that the adaptive Stroud SCM is more efficient and the statistical moments of the hydraulic head can be more accurately estimated.

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