Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport (Communications in Computational Physics, 2009)
Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developedexact first and second nonlocal moment equations for advective-dispersive transportin finite, randomly heterogeneous geologic media. The velocity and concentrationin these equations are generally nonstationary due to trends in heterogeneity, conditioningon site data and the influence of forcing terms. Morales-Casique et al. (Adv.Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of theseequations recursively to second order in the standard deviation σY of (natural) log hydraulicconductivity, and iteratively to higher-order, by finite elements followed bynumerical inversion of the Laplace transform. They did the same for a space-localizedversion of the mean transport equation. Here we recount briefly their theory and algorithms;compare the numerical performance of the Laplace-transform finite elementscheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled withan alternating split operator approach; and review some computational results due toMorales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light onthe accuracy and computational efficiency of their recursive and iterative solutions incomparison to conditional Monte Carlo simulations in two spatial dimensions.