Volume 18, Issue 3

Abstract. In this study, we present the basic-concepts of ground-water hydraulic on stochastic media, whose the Theis equation is used in the transient movement of groundwater as a result of pumping in a confined aquifer in saturated porous media under random parameters. A special importance is given on circumstances which requires a high-dimension stochastic to obtain a certain precision in probability space. As an alternative, we introduce quasi-Monte Carlo methods considering Sobol and Halton sequences for uncertainty assessment. Accuracy and efficiency are studied here with allusion to model problem using as reference the solution obtained by Monte Carlo method and numerical experiments on two-dimensional random field identified some restrictions with the increase of realizations number. To represent the realizations of the hydraulic conductivity field based on available data was used the Karhunen–Loève expansion.

Abstract. We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

Abstract. In this paper, we present the application of our recently developed Green's function Monte Carlo algorithm to the solution of the one-dimensional Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional Green's function Monte Carlo approach for the solution of partial differential equations subjected to Neumann and mixed boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. Our algorithm, motivated by the work of K. K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. In the past, we have applied it to the solution of the one-dimensional Laplace equation and the modified Helmholtz equation. In this work, we apply it to the solution of the Helmholtz equation. In the case of the Helmholtz equation, unlike the Laplace equation and modified Helmholtz equation, the algorithm is constrained to quarter-wavelength length scales, a constraint that is the result of resonance in the Green's function for the Helmholtz equation. This constraint is also present in the case of the Helmholtz equation subjected to Dirichlet conditions and is not specific to Neumann and mixed boundary conditions. However, within this constraint, excellent agreement has been obtained between an analytical solution and numerical results.