| Issue |
ESAIM: M2AN
Volume 56, Number 1, January-February 2022
|
|
|---|---|---|
| Page(s) | 317 - 347 | |
| DOI | https://doi.org/10.1051/m2an/2021087 | |
| Published online | 10 February 2022 | |
Exact solution and the multidimensional Godunov scheme for the acoustic equations
Institute for Mathematics, Würzburg University, Emil-Fischer-Straße 40, 97074 Würzburg, Germany
* Corresponding author: w.barsukow@mathematik.uni-wuerzburg.de
Received:
7
September
2018
Accepted:
20
December
2021
The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.
Mathematics Subject Classification: 35B08 / 35B44 / 35C05 / 35E05 / 35E15 / 35F40 / 35L03 / 35Q35 / 65M08
Key words: Godunov scheme / multidimensional Riemann Problem / acoustic equations / linearized Euler equations
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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