Eulerian-Lagrangian Localized Adjoint Methods for Nonlinear Reactive Transport

We consider Eulerian-Lagrangian Localized Adjoint Methods (ELLAMs) applied to nonlinear model equations governing solute transport and sorption reactions in porous media. Solute transport in the aqueous phase is modeled by standard advection and diffusion processes while sorption reactions are modeled with a local equilibrium assumption and the nonlinear Freundlich equilibrium isotherm. In this work we focus particularly on the case when the Freundlich isotherm is not Lipschitz continuous. The nonlinear parabolic equation that results can yield solutions with self-sharpening fronts under certain choices of boundary and initial data. ELLAMs were designed to circumvent the stability and accuracy limitations of standard methods when applied to advection-dominated transport equations with sharp fronts, and are thus a natural choice for this application. Indeed, for a wide range of problems in subsurface flow and transport, ELLAMs have produced accurate, non-oscillatory results for significantly larger time steps and coarser grids than is possible with standard methods such as Galerkin finite elements. The particular form of the nonlinearity in the reactive transport model we study presents a number of difficulties, which we address in the context of ELLAMs. We will present our implementations of ELLAMs to the reactive transport model as well as numerical comparisons to standard Galerkin finite elements and finite difference solutions.