Design and Validation of a Discontinuous Galerkin Baroclinic Ocean Model Using Prismatic Elements

Over the last decade, increasing efforts have been directed towards the development of marine models using unstructured meshes. Such models allow for an accurate representation of the coastlines (e.g., islands, narrow straits) and the bathymetry. The mesh can easily be refined in regions of interest or coarsened in those regions where the dynamics is less demanding. Finally, unstructured meshes set up in spherical geometry allow to avoid singularities at the poles, rendering those techniques potentially useful for global ocean modeling.
We present the current version of the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM, http://www.climate.be/SLIM). We solve three-dimensional baroclinic flows with equal order discontinuous our non-conforming finite elements. Emphasis is put on several key choices in the development of the model. First, the time stepping procedure is analyzed. A so-called implicit-explicit (IMEX) procedure is used. To allow flexible time step modification, multistep methods are avoided.
Multistage Runge-Kutta schemes appear to be very attractive. As surface gravity waves are known to be by far the fastest phenomenon, it is usual to treat them implicitly to keep large time steps. We evaluate a mode splitting approach that uses different time stepping schemes for the three-dimensional phenomena, and their two-dimensional horizontal restriction. Therefore, the linear system for the three-dimensional equation is typically block diagonal. Such a system can be easily solved column by column and does not imply the allocation of a global matrix. This splitting procedure is innovative, because it is designed so that it can be viewed as a specific time stepping scheme, rather than a spatial approximation. Further, we present the critical aspects of the discretization of spatial operators using the Discontinuous Galerkin method. We highlight the application of the model to baroclinic test cases.