Stability of difference approximations to certain partial differential equations of fluid dynamics

Abstract

First- and second-order difference approximations to certain partial differential equations of fluid dynamics are investigated for the von Neumann necessary condition for stability. The nonsteady equations examined consist of advection, diffusion, and inertial terms. Although the general equation considered represents no particular atmospheric process, it does have features found in many meteorological problems. Five approximations to the diffusion equation are studied, three of which are shown to be stable. Two of the three approximations to the advection-diffusion equation investigated are found to be stable. Twelve approximations to the advection-inertial equation are examined; eight are found to be stable and two are found to be slightly unstable. One two-step scheme, each step of which is individually stable, is shown to be unstable. For the advection-diffusion-inertial equation, 2 two-step schemes are formed from the preceding stable schemes. The analysis shows that such combinations are not necessarily stable.