Discretization of a model for the formation of longshore sand ridges

This paper presents and evaluates the numerical solution of a coupled system of equations that arises in a model for the formation and evolution of three-dimensional longshore sand ridges. The model is based on the interaction between surficial or internal weakly nonlinear shallow-water waves, having weak spanwise spatial dependence, and the deformable bottom topography. The presentation of the details concerning the discretization of the model is primarily motivated by: (1) the model involves equations for which little is known regarding its solutions; (2) we believe that the methodology used in simplifying the solution to the coupled sand ridge model may be of interest to other researchers in the geophysical community; and (3) the predictor-corrector scheme presented here, which combines finite difference techniques and fixed-point methods, is simple, fast, and general enough to be used in the discretization of other partial differential equations with local nonlinearities whose solutions are smooth and bounded.