The Performance of Classical versus Modern Finite-Volume Advection Schemes for Atmospheric Modeling in a One-Dimensional Test-bed

Abstract

The numerical solution of the transport (i.e., the continuity) equation for trace species, particularly in three-dimensional circulation models, has recently received great attention. Two different approaches are common. First, the classical numerical methods employed in circulation models for the solution of other continuity equations are used for the transport problem. In these methods, filters are commonly applied to control undesirable features in the solution. Second, methods were developed that were specially designed to obviate the problems arising when the transport equation is numerically solved. Here, both approaches are investigated and compared in a simple one-dimensional test-bed. The methods discussed encompass leapfrog time stepping, with both discrete and spectral spatial resolution, various filters, and four Eulerian finite-volume advection schemes: the Prather scheme, the Bott scheme, the piecewise parabolic method (PPM), and four versions of the multidimensional positive-definite advection transport algorithm (MPDATA). The focus of the discussion is on computational cost and on undesirable numerical artifacts, such as dispersive ripples, negative concentrations, phase errors, and numerical diffusion. An appropriate transport scheme should totally avoid or strongly control all of these artifacts. It is questionable whether classical numerical methods are able to meet this requirement. The modern advection schemes, in contrast, perform considerably better, where computationally relatively inexpensive schemes (Bott, MPDATA) are recommended if a certain amount of numerical diffusion can be tolerated. The latter problem can almost completely be avoided if more sophisticated methods (Prather, PPM) are employed. This, however, increases the computational effort considerably.