A moving grid Eulerian Lagrangian localized adjoint method for solving one-dimensional nonlinear advection-diffusion-reaction equations

We develop a moving grid Eulerian-Lagrangian Localized Adjoint Method (ELLAM) to solve the one-dimensional nonlinear Advective-Dispersive-Reactive Equation (ADRE) that governs contaminant transfer with biological/chemical/radioactive processes. The method uses a moving grid to define the solution and test functions. Consequently, the need for spatial interpolation, which is a major issue in conventional ELLAM formulations, is avoided. We apply the developed method for variable-coefficient ADRE with different time and spatial discretizations. Numerical experiments show the strength of the developed scheme. Accurate results are obtained even when using large time steps. In the last part of this paper, ELLAM is used to simulate the nonlinear ADRE that arises for reactive subsurface transport with biodegradation.