Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

Abstract

Abstract. In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, 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 phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M −2 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 confirmed by numerical experiments.

A low-Mach Roe-type solver for the Euler equations allowing for gravity source terms

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Numerical dissipation of upwind schemes in low Mach flow

“…Hence, the form of this stabilization does not necessarily require a physical basis, but it must not dominate the flow physics (as happens with the standard flux at low Mach). This modification also allows good stability according to the standard CFL condition, as opposed to standard preconditioned methods where stability in explicit time stepping is prohibitive [5], thus can be used where the time stepping is not constrained by the low Mach portion of the flow. In addition, it preserves exactly a stationary material interface.…”