Modern Finite-Volume Advection Schemes: Numerical Experiments in a One-Dimensional Test-Bed

Abstract

The numerical solution of the transport (i.e. the continuity) equation for trace species, in particular in three-dimensional circulation models, has recently received much attention. Two conceptually different approaches are common. Firstly, classical numerical methods employed in circulation models for the solution of other continuity equations are also used for the. transport problem. Secondly, methods have been developed, that are especially designed to obviate problems arising in the numerical solution of the transport equation. In a recent study (Muller [1]) both approaches were investigated and compared in a simple one-dimensional test-bed. It was concluded that it is questionable whether classical numerical methods are able to control undesirable numerical artefacts such as dispersive ripples, negative concentrations, phase errors and numerical diffusion. The modern advection schemes, in contrast, perform considerably better, where computationaly relatively inexpensive schemes (Bott [2], Smolarkiewicz, and Grabowski [3]) are preferable if a certain amount of numerical diffusion can be tolerated. Moreover, numerical diffusion can be controled much better if more sophisticated methods (Prather [4], Carpenter et al. [5]) are employed. This, however, increases the computational effort considerably. Here, some hitherto undis-cussed aspects of Eulerian finite-volume schemes are elucidated. In particular, it is shown that interpolation based schemes can advect multiple peak structures properly, provided the resolution is sufficient. Further, advection schemes, originally designed for transporting shock fronts, are shown to possess problematic features if applied to other (in this case triangular) structures.