Really TVD advection schemes for shelf seas

During the last decade large efforts have been devoted to the development of high-resolution schemes to solve advection problems. High-resolution conservative numerical schemes satisfying conservative, monotonicity preserving and shock-capturing properties are nowadays widely used in ocean modeling. Among these, TVD schemes, based on the concept of Total Variation Diminishing (TVD), were progressively adopted because of their good behavior that guarantees a solution free from numerical artifacts (no overshooting, no spurious oscillation, small diffusion) that can spoil the physical significance of the results.

Most of the TVD schemes and associated limiters have been originally developed in idealized one-dimensional flows described by a linear advection. In finite volume marine models, one has however often to deal with the depth integrated advection equation. This formulation is usually preferred because of its conservative form that is particularly suited to numerical treatment using a finite volume approach. Conservative numerical schemes can be easily formulated to ensure that the total mass of the advected quantity is conserved. This property is very valuable in the context of environmental studies for which a strict equilibrium of the mass budget of pollutants is often more relevant that the raw accuracy of the integration scheme. In the same context, the numerical scheme should also produce neither new local extremum nor negative concentrations, i.e. it should be monotonicity preserving which is implied by the TVD property.

The development of TVD schemes for the resolution of advection equations written in the conservative form is however not trivial. Numerical experiments show that the blind application to the depth-integrated equation of the usual TVD schemes and associated flux limiters introduced in the context of linear advection can lead to non-TVD solutions in presence of complex geometries. Spatial and/or temporal variations of the local bathymetry can indeed break the TVD property of the usual schemes. Really TVD schemes can be recovered by taking into account the local depth and its variations in the formulation of the flux limiters. Using this approach, a generalized superbee limiter is introduced and validated.