Efficient Time Steppers for Ocean Modeling

The development of suitable and fast time integration methods for ocean modeling constitutes an important challenge. No single time-discretization works well for all physical processes in a complex marine model, as different subsystems have widely different characteristic time scales. We believe that building appropriate time stepping strategies for multi-scale computations will enable us to gain an order of magnitude. In the framework of the Second-generation Louvain-la-Neuve Ice-ocean model (SLIM, www.climate.be/SLIM), a discontinuous Galerkin finite element method based on unstructured meshes is used for the spatial representation. It is therefore well suited for simulating estuarine and coastal flows where capturing complex topography is crucial. Moreover, unstructured grids also allow to capture a wide spectrum of time and length scales in a single model since the spatial resolution can be increased in regions of interest. The meshes are built by means of the open source software GMSH (www.geuz.org/gmsh). Unstructured-mesh generation processes are complex and, even though it is possible to control average element sizes in specific regions of the domain, it is not the case for each element size. The smallest element is usually much more smaller than the criterion that was prescribed a priori and it determines the stable time step for the entire model. Therefore, the computational efficiency of explicit time-stepping methods may be drastically low. However, explicit time stepping schemes are still very attractive. Multirate schemes present a way to partly circumvent the stability restrictions by gathering the mesh elements in groups that satisfy the local CFL stability conditions for a certain range of time steps. But the transitions between these multirate groups have to be accommodated in order to ensure a coherent communication. These methods turn out to be well suited and optimized for the discontinuous Galerkin meshes and may dramatically reduce the total computational efforts for large-scale applications. In the context of multi-scale marine modeling, the application of the multirate approach is not limited to the hydrodynamics but also takes into account other hydrological processes such as biological and passive tracers. Moreover, considering wetting and drying areas is also a quite critical challenge for a multirate time stepper.