Influence oflow Mach numbers on thenumerical dissipation of Godunov-typeschemes: a comparison between Roeand HLLscheme

Upwind schemes for the Euler equations face three kinds of problems in the low Mach number regime: The stiffness due to the presence of fast acoustic and slow entropy and shear waves can be overcome – at least for steady problems – by preconditioning the physical equations, see for example 1, 2]. Secondly, the ??(M²)-pressure variations get lost in the O(1)-global pressure due to finite precision arithmetics. This cancellation problem was dealt with in 3]. The third problem originates in the numerical viscosity of the upwind schemes and is – in the author's view of the matter – not fully understood to date. Asymptotic analyses of the upwind schemes such as 4] suggest that the pressure field will completely degenerate, producing variations of the wrong order of magnitude. Our numerical andanalytical results give a more precise picture of the problem and are – at least in parts – contradicting the established view. We will prove in this paper that complete Riemann solvers such as Roe's behave completely different to incomplete Riemann solvers such as HLL in the low Mach number regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)