Stability of preconditioned finite volume schemes at low Mach numbers

Abstract

Abstract. In [4], Guillard and Viozat propose a nite volume method for the simulation of inviscid steady as well as unsteady ows at low Mach numbers, based on a preconditioning technique. The scheme satises the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modication of the dissipation term within the numerical ux function. Unfortu-nately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step t does not satisfy the requirement to be O(M 2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to t = O(M), M! 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phe-nomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical ux function possesses an eigenvalue growing like M2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards conrmed by numerical experiments