D. Y. Le Roux

Advection schemes for unstructured grid ocean modelling

We study advection schemes for unstructured grid ocean models. Four linear advection schemes are investigated by solving a scalar transport equation. Schemes under consideration include continuous, nonconforming and discontinuous finite elements and finite volumes. A comprehensive derivation of the numerical schemes is presented and conservation and dispersion properties are discussed. An assessment is made by performing the test problem introduced by Hecht et al. [J. Geophys. Res. 100 (1995) 20763] in which a passive scalar field is advected through an analytical Stommel gyre. It is found that continuous finite elements and finite volumes have some difficulties to represent accurately solutions with steep gradients. As a result they are prone to generate unphysical oscillations. On the other hand, discontinuous and non-conforming finite element schemes perform better. This is due to their higher flexibility that makes them better suited to highly sheared flows

Conservative semi-implicit semi-Lagrangian scheme for simulation of shallow flows

A fully conservative semi-Lagrangian (SL) scheme is presented to solve for the shallow-water equations. Existing inherently conservative SL schemes only ensure the conservation of mass while momentum is not fully conserved. The gravity terms, which are mainly responsible for the wave structure in dam break flows, are then discretized by using traditional non-conservative Eulerian schemes. In the presence of large variations in water surface (e.g., dam-break type flows), such an approach leads to incorrect shock speed and highly oscillatory results. Indeed, if the conservation of the gravity terms is forced, the use of existing schemes will be restricted to small time steps. In this paper we present a fully conservative scheme which can accurately simulate the shallow flows with a large time step. In our approach, both convective and gravitational terms are treated in a conservative manner, which ensures an accurate shock speed. The fully conservation property improves considerably the performance of common SL schemes for a wide range of practical applications.