Commun. Comput. Phys.,
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Volume 4.


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

Yan Ding 1, Tiejun Li 1*, Dongxiao Zhang 2, Pingwen Zhang 1

1 LMAM and School of Mathematical Sciences, Peking University, Beijing 100875, P.R. China.
2 Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China; and Department of Civil and Environmental Engineering, and Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089, USA.

Received 5 May 2007; Accepted (in revised version) 8 December 2007
Available online 27 February 2008

Abstract

In this paper we develop a Stochastic Collocation Method (SCM) for flow in randomly heterogeneous porous media. At first, the Karhunen-Lo\`{e}ve 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 variables. 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 {\it adaptive 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.

AMS subject classifications: 60H15, 65M70, 76M22, 76S05

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Key words: Adaptive Stroud stochastic collocation method, Karhunen-Lo\`{e}ve expansion, Monte Carlo simulation, random porous flow.

*Corresponding author.
Email: dyan@math.pku.edu.cn (Y. Ding), tieli@pku.edu.cn (T. Li), donzhang@usc.edu (D. Zhang), pzhang@pku.edu.cn (P. Zhang)
 

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