Coastal ocean models: two useful finite element methods

Methods for solving computational fluid dynamic problems in shallow water have changed dramatically over the last three decades, along with fundamental changes in our understanding of numerical methods. The early search for stable and accurate techniques was driven by the presence of spurious computational modes in sea level and/or velocity for most obvious choices of computational grids. Building on collective experiences, newer methods have overcome many of these earlier problems with spurious modes, advection instabilities, accuracy, and efficiency. Two classes of models are described here: the first solves a modified form of the governing equations (wave equation) with simple finite elements, and the other solves the primitive equations with finite elements that contain no spurious modes. The models are applied to case studies of tides around New Zealand, to advection controlled flows during a flood, and to tsunami generated by a submarine landslide on the continental shelf. Results from these studies are summarized.