On Forward-in-Time Differencing for Fluids

Abstract

This note discusses the extension of the dissipative advection schemes, often referred to in meteorological literature as Crowley-type schemes, on advection equations with arbitrary forcing and/or source terms included. Since such equations constitute a prototype of prognostic equations for fluids, the considerations herein are relevant to a variety of atmospheric problems. The thesis of this note is that, no matter how accurate the advection scheme employed, the entire equation is approximated to, at most, O(?t), which is a consequence of disregarding forcing terms in the derivation of Crowley-type schemes. The consequences of this truncation error may be quite severe depending on the particular problem at hand. The remedy proposed is simple and easy to implement in any numerical model using forward-in-time differencing. Theoretical considerations are illustrated with an example of a flow of the density-stratified fluid past a two-dimensional mountain.