Document Type : Original Research Paper
Authors
Department of Rock Mechanics, Faculty of Mining and Materials Engineering, Tarbiat Modares University, Tehran, Iran
Abstract
Three numerical schemes for groundwater contaminant transport are compared within a single three-dimensional implementation: the finite difference method (FDM), the Galerkin finite element method (GFEM) and a classical upwind Petrov–Galerkin finite element method (PGFEM). All schemes use the same mesh, source representation, boundary conditions and time integrator. Numerical verification is performed against the Ogata–Banks and Wexler analytical solutions. Performance depends primarily on the grid Péclet number (Pe_g). The observed orders of convergence are 1.75, 1.80 and 0.91 for GFEM, PGFEM and upwind FDM, respectively. With central differencing, FDM converges at an order of 1.71 and is identical to lumped-mass GFEM in one-dimensional problems with no transverse variation. For Pe_g ≲ 2, GFEM is more accurate and remains monotone along the flow direction, so upwind weighting introduces unnecessary artificial dispersion. For Pe_g ≳ 4, GFEM develops oscillations; at Pe_g = 100, it overshoots the source concentration by 5.2% and becomes negative ahead of the front. PGFEM remains monotone along the flow direction but has twice the Galerkin error, with artificial dispersion 49 times the physical value. Both finite element schemes also exhibit a separate undershoot associated with violation of the discrete maximum principle. Neither mesh refinement nor upwinding eliminates this effect. The Domenico approximation underestimates the exact solution by 44–73% ahead of the front. In the three-dimensional patch-source problem, convergence is governed by transverse source resolution rather than longitudinal mesh spacing.
Keywords
- Petrov–Galerkin finite element method
- Galerkin finite element method
- finite difference method
- contaminant transport
- groundwater
Main Subjects
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