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Stationarity preserving schemes for multi-dimensional linear systems

Author: Wasilij Barsukow
Journal: Math. Comp. 88 (2019), 1621-1645
MSC (2010): Primary 35L40, 65M06, 65M08, 39A70
Published electronically: November 13, 2018
MathSciNet review: 3925479
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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).

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Additional Information

Wasilij Barsukow
Affiliation: Institute for Mathematics, Würzburg University, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

Keywords: Stationarity preserving, linear acoustics, system wave equation, vorticity preserving
Received by editor(s): December 4, 2017
Received by editor(s) in revised form: May 23, 2018
Published electronically: November 13, 2018
Additional Notes: The author was supported by the German National Academic Foundation.
Article copyright: © Copyright 2018 Wasilij Barsukow