Eulerian-Lagrangian localized adjoint methods for systems of nonlinear advective-diffusive-reactive transport equations

Abstract

The contamination of groundwater by various hazardous materials has emerged as a primary environmental issue. The pollution of oil reservoirs is a closely related problem in that microorganisms are involved in the contaminant process. The mathematical models that describe these phenomena involve a set of nonlinear advective-diffusive-reactive transport equations, which may involve reactions with all the species and are themselves coupled to growth equations for the subsurface bacterial population. In this article, we discuss and compare different mathematical models, present Eulerian-Lagrangian localized adjoint methods (ELLAM) and combine them with specific linearization techniques to solve these nonlinear transport systems. The derived numerical schemes systematically adapt to the changing features of governing equations. The relative importance of advection, diffusion and reaction is directly incorporated into the schemes by judicious choice of the test functions in the variational formulations. Numerical experiments are presented to show the potential of these methods.