Streamline Upwind Petrov-Galerkin–Based Shallow Water Model for Large-Scale Geophysical Flows in Cartesian and Spherical Coordinates

Abstract

The development and implementation of a stabilized finite-element model for the simulation of large-scale geophysical flows using the shallow water equations (SWEs) is presented. The model is derived from the mass and momentum conservative forms of the SWE, with wetting–drying implemented using a front tracking algorithm. Transient hydrodynamic phenomena are resolved using run time h-mesh adaption, such that the initial grid resolution only needs to capture bathymetric features. Cartographic mapping is used to allow the use of Cartesian master elements when the meshing is performed in spherical coordinates. The model is validated using five applications designed to test mass conservation and robustness of the wet–dry scheme, and the cartographic mapping is implemented. The presented finite-element model numerical scheme for large-scale geophysical flows overcomes the limitations of mass conservation that have plagued older finite-element model schemes dependent on the generalized wave continuity equation (GWCE). The presented numerical model derives its novelty from the combination of a mass and momentum conservative finite-element model framework, true wetting–drying, and implicit time stepping with a spatially adaptive mesh and temporally adaptive time integration scheme. To the author’s knowledge, no prior finite-element model works exist with this combination of features in the realm of shallow water simulations for large-scale geophysical flows.