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 surrcial 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-nite diierence techniques and xed-point methods, is simple, fast, and general enough to be used in the discretization of other partial diierential equations with local non-linearities whose solutions are smooth and bounded.