Numerical simulation of one-dimensional and two-dimensional mass transport in groundwater, using Galerkin, Petrov-Galerkin and localized adjoint methods.

Abstract

Advection-diffusion transport equations are important in many branches of engineering and applied science. These equations are characterized by a nondissipative (hyperbolic) advective transport component and a dissipative (parabolic) diffusive component. When diffusion is the dominant process, most finite element numerical solution procedures perform well. However, when advection is the dominant transport process, most finite element numerical procedures exhibit some combination of excessive nonphysical oscillations and excessive numerical diffusion. Many numerical methods use Eulerian analysis to solve the advective diffusion equation. Some of them suffer from accuracy limitations, and others suffer from strict Courant number limitation. A Localized Adjoint Petrov-Galerkin Method is proposed to solve the multi-dimensional advection-diffusion equation. The method uses a weight function that is a numerical solution of adjoint state equations on a sequence of nested grids. Solutions corresponding to a given member of the sequence are ideally suited for parallel computation. One-dimensional numerical results are presented. The numerical results demonstrate that the method is accurate for low Peclet numbers. For large Peclet numbers numerical dispersion is introduced in the concentration profiles. Another method, the incomplete cubic Hermitian Galerkin, is presented to solve the transport equation for high Peclet number. One- and two-dimensional numerical results are presented. The numerical results demonstrate less oscillation and dispersion than obtained by linear Galerkin method.