Comparison of advection schemes for high-order h-p finite element and finite volume methods

We review and compare advection schemes designed for high-order finite element/finite volume methods. The emphasis is on studying, by numerical examples, the properties of these schemes in terms of accuracy, and monotonicity, and their viability for oceanic applications. The schemes reviewed are classical spectral element, Taylor Galerkin Least Square method, the Discontinuous Galerkin method and high-order finite volume method. The latter two schemes exhibit a definite robustness due to their small, but finite, inherent numerical dissipation. They also prove the most flexible since their discontinuous representation of the solution allows easy implementations of flux limiting or adaptive procedure. Finally, an ad-hoc but simple adaptive procedure is presented to illustrate DGM's potential; this procedure proved to be extremely effective at controlling Gibbs oscillations in 1D but was too dissipative on the Hecht problem.