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The acoustic equations derived as a linearization of the Euler equations area valuable system for studies of multi-dimensional solutions. Additionally theypossess a low Mach number limit analogous to that of the Euler equations.Aiming at understanding the behaviour of the multi-dimensional Godunov schemein this limit, first the exact solution of the corresponding Cauchy problem inthree spatial dimensions is derived. The appearance of logarithmicsingularities in the exact solution of the 4-quadrant Riemann Problem in twodimensions is discussed. The solution formulae are then used to obtain themultidimensional Godunov finite volume scheme in two dimensions. It is shown tobe superior to the dimensionally split upwind/Roe scheme concerning its domainof stability and ability to resolve multi-dimensional Riemann problems. It isshown experimentally and theoretically that despite taking into accountmulti-dimensional information it is, however, not able to resolve the low Machnumber limit.

Exact solution and the multidimensional Godunov scheme for the acoustic equations