Exact solution and the multidimensional Godunov scheme for the acoustic equations

TL;DR

This paper derives the exact solution for the 3D acoustic equations, develops a multidimensional Godunov scheme, and compares its performance to other schemes, highlighting its advantages and limitations in low Mach number regimes.

Contribution

It provides the first exact 3D solution for the acoustic equations and constructs a multidimensional Godunov scheme with improved stability and resolution capabilities.

Findings

The exact 3D solution reveals logarithmic singularities in 2D Riemann problems.

The multidimensional Godunov scheme outperforms dimensionally split schemes in stability.

The scheme cannot resolve the low Mach number limit despite multi-dimensional considerations.

Abstract

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.