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.
Highlights
Hyperbolic systems of PDEs in multiple spatial dimensions exhibit a richer phenomenology than their onedimensional counterparts
There is a number of striking differences to the one-dimensional case that appear in multiple spatial dimensions
This paper presents an exact solution of the acoustic equations in three spatial dimensions
Summary
Hyperbolic systems of PDEs in multiple spatial dimensions exhibit a richer phenomenology than their onedimensional counterparts. Whereas the solution to advection in multiple dimensions is not very different from its one-dimensional counterpart, acoustics exhibits a number of new features This has led [1, 2, 4, 5, 12, 35, 38, 41] and others to studies of the linearized acoustic operator. This paper studies a multi-dimensional Godunov scheme excluding both dimensional splitting and approximate evolution as reasons for the failure to resolve the low Mach number limit. This paper for the first time gives a detailed derivation of the distributional solution to the Cauchy problem of linear acoustics, without the restriction to smooth initial data. The Godunov scheme is derived using the framework of distributional solutions, as the initial data of a Riemann Problem are discontinuous. The ability of the method to resolve the low Mach number limit is studied both theoretically and experimentally there as well
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