A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields (Communications in Computational Physics, 2009)
Stochastic collocation methods as a promising approach for solving stochasticpartial differential equations have been developed rapidly in recent years. Similarto Monte Carlo methods, the stochastic collocation methods are non-intrusive in thatthey can be implemented via repetitive execution of an existing deterministic solverwithout modifying it. The choice of collocation points leads to a variety of stochasticcollocation methods including tensor product method, Smolyak method, Stroud 2 or 3cubature method, and adaptive Stroud method. Another type of collocation method,the probabilistic collocation method (PCM), has also been proposed and applied toflow in porous media. In this paper, we discuss these methods in terms of their accuracy,efficiency, and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variabilityand correlation length. This study reveals that the Smolyak method and the PCMoutperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial rolein the performance of the stochastic collocation methods. Our numerical experimentsindicate that the required random dimensionality increases slightly with the decreaseof correlation scale and moderately from one to multiple physical dimensions.