Modeling two-dimensional reactive transport using a Godunov-mixed finite element method

SummaryThe development of a model to simulate transport of materials in variable-depth flows is discussed. The model numerically approximates solutions to the advection–dispersion-reaction equation using a time-splitting technique where the advective, dispersive, and reactive parts of the equation are solved separately. An explicit finite-volume Godunov method is used to approximate the advective part while a hybridized mixed finite element method is used to solve for the dispersive step. A backward Euler method is used to solve the reactive component. Rather than solving each component once at each time step, the advective and reactive steps are fractionally and symmetrically split around the dispersive step, so that half of a reactive and advective step are solved before and after each dispersive step. Since the dispersive step is implicit, but computationally expensive, while the advective step is explicit but has time step constraints, this allows stable and more efficient schemes to be implemented in contrast to non-split or simple time-split algorithms. This technique allows problems with high grid Peclet numbers, such as transport problems with sharp solute fronts, to be solved without oscillations in the solution and with virtually no artificial diffusion. By applying the technique to variable depth flows, a variety of applications to transport and reaction problems in surface water and unconfined aquifers can be undertaken. Numerical results for several non-reactive and reactive transport problems in one- and two-dimensions are presented. Observed convergence rates are up to second-order for these simulations.