Stationarity preserving schemes for multi-dimensional linear systems

Abstract

There is a qualitative difference between one-dimensional and multi-dimensional solutions to the Euler equations: new features that arise are vorticity and a nontrivial incompressible (low Mach number) limit. They present challenges to finite volume methods. It seems that an important step in this direction is to first study the new features for the multi-dimensional acoustic equations. There exists an analogue of the low Mach number limit for this system and its vorticity is stationary. It is shown that a scheme that possesses a stationary discrete vorticity (vorticity preserving) also has stationary states that are discretizations of all the analytic stationary states. This property is termed stationarity preserving. Both these features are not generically fulfilled by finite volume schemes; in this paper a condition is derived that determines whether a scheme is stationarity preserving (or, equivalently, vorticity preserving) on a Cartesian grid. Additionally, this paper also uncovers a previously unknown connection to schemes that comply with the low Mach number limit. Truly multi-dimensional schemes are found to arise naturally and it is shown that a multi-dimensional discrete divergence previously discussed in the literature is the only possible stationarity preserving one (in a certain class).