A generalized, frequency–independent, finite–element method is proposed for the numerical simulation of combined diffraction, refraction, radiation, reflection and absorption of water waves. Plane waves are used as an external function space, exploiting the standard set of approximation functions defined by the finite element method to enforce conformity and to impose boundary conditions in a straightforward manner by applying the method to the plane waves on each element.

A theoretical formulation and corresponding numerical solutions are presented for fluid flow and sediment transport past evolutionary sand dunes. Time-dependent curvilinear coordinates are employed to fully couple flow aloft with the developing landform. The differential conservation law that defines shape of the lower boundary depends on details of local surface stress, thereby favoring the large eddy simulation of the boundary layer. To shrink the gap between the time scales characteristic of planetary boundary layer flows $O\left({10}^3\right)\text{s}$ and sand dune evolution $O\left({10}^6\right)\text{s}$, a hypothetical “severe-wind scenario” is adopted with the saltation flux amplified up to 3 orders of magnitude. While the results are largely insensitive to the rescaling, the efficacy of computations is greatly improved. The flux-form partial differential equation for the interface profile—via saltation and sand avalanches—is formulated as an advection-diffusion equation, to facilitate discrete integrations. Numerical experiments verify the adopted theoretical framework by reproducing scaling results reported in the literature. The versatility of the approach is illustrated with evolution of a sandhole—an example of application likely never addressed in the literature, yet realizable in nature.