The first ocean general circulation models developed in the late sixties were based on finite differences schemes on structured grids.
Many improvements in the fields of engineering have been achieved since three decades with the developments of new numerical methods based on unstructured meshes. New second generation models are now under study, with the aim of taking advantage of the potential of modern numerical techniques such as finite elements on unstructured grids. A popular new trend in engineering applications is the high- order Discontinuous Galerkin method, presenting many interesting numerical properties in terms of dispersion and dissipation, errors convergence rates, advection schemes, mesh adaptation.
The key issue for unstructured high-order finite elements approach is the computational cost of the calculations. Therefore, we analyze three routes to have new efficient schemes
(1) Exploit single precision BLAS/LAPACK for the efficient implementation of the explicit and implicit discontinuous Galerkin methods.
(2) Use novel time-integration procedures for multi-scale models and adpat the time-integration scheme to the physical processes.
(3) Introduce multi-level methods for the implicit linear and non- linear solvers with multigrid methods as a preconditioner for stiff, non-linear and non-positive-definite systems.
Each route could reduce the computational cost by one order of magnitude.