An adaptive multi-dimensional Eulerian-Lagrangian finite element method for simulating advection-dispersion.

Abstract

Advection-dispersion is generally solved numerically with methods that treat the problem from one of three perspectives. These are described as the Eulerian reference, the Lagrangian reference or a combination of the two that will be referred to as Eulerian-Lagrangian. Methods that use the Eulerian-Lagrangian approach incorporate the computational power of the Lagrangian treatment of advection with the simplicity of the fixed Eulerian grid. A modified version of a relatively new adaptive Eulerian-Lagrangian finite element method is presented for the simulation of advection-dispersion. Advection is solved by an adaptive technique that automatically chooses a local solution technique based upon a criterion involving the spatial variation of the gradient of the concentration. Moving particles (the method of characteristics; MOC) are used to define the concentration field in areas with significant variation of the concentration gradient. A modified method of characteristics (MMOC) called single-step reverse particle tracking is used to treat advection in areas with fairly uniform concentration gradients. As the simulation proceeds, the adaptive technique, as needed to maintain solution accuracy and optimal simulation efficiency, adjusts the advection solution process by inserting and deleting moving particles to shift between MMOC and MOC. Dispersion is simulated by a finite element formulation that involves only symmetric and diagonal matrices. Despite evidence from other investigators that diagonalization of the mass matrix may lead to poor solutions to advection-dispersion problems, this method seems to allow "lumping" of the mass matrix by essentially decoupling advection and dispersion. Based on tests of problems with analytical solutions, the method seems capable of reliably simulating the entire range of Peclet numbers with Courant numbers that range to 15.