Efficient Parallel Multirate Time Stepping with Application to the World Ocean.

The development of suitable and fast time integration methods for ocean modeling constitutes an important challenge. No single time-discretisation works well for all physical processes in a complex marine model, as different subsystems have widely different characteristics in terms of time scales, dynamic behaviour, and accuracy requirements. We believe that building appropriate time stepping strategies for multi-scale computations will enable us to gain an order of magnitude. Indeed, 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. Multirate schemes represent a class of methods that use various time steps on different grid cells. The strategy consists in splitting the domain in a smart way. Grid cells are gathered in different groups that satisfy the local CFL stability conditions for a certain range of time steps. Standard explicit RungeKutta methods are applied on independent partitions while buffer groups have to be introduced between them, with adapted methods, in order to accommodate the transitions between them. These methods are especially suited for the Discontinuous Galerkin spatial discretization. Nevertheless, development of such methods is still challenging. Both, stability requirements and conservation properties should be satisfied. Two approaches are explored. Constantinescu introduced a conservative 2nd order scheme while Schlegel proposed a 3rd method that is, unfortunately, not conservative. Large-scale applications like the Great Barrier Reef require the use of parallel computers. Some kind of load balancing strategy has to be supplied to accomodate multirate schemes: indeed, small elements have a higher cost than large elements in such a strategy. Moreover, small elements at inter-processor interfaces will require more frequent updates. The key idea consists in creating an optimized mesh partition in a way that the amount of grid cells of the different multirate groups is ideally the same on each computer core. However, a compromise should also be found between the effective work on each processor and the amount of communications between them