Flux Splitting for stiff equations: A notion on stability

TL;DR

This paper investigates the stability of IMEX schemes for low Mach number flows, deriving criteria that ensure uniform stability regardless of Mach number, and introduces a new flux splitting method based on characteristic decomposition.

Contribution

It provides a stability analysis for IMEX schemes in low Mach regimes and proposes a novel flux splitting approach that maintains stability independent of Mach number.

Findings

Derived a stability criterion for linear hyperbolic systems.

Identified that characteristic-based flux splitting avoids time step deterioration.

Demonstrated the effectiveness of the new splitting method in stability analysis.

Abstract

For low Mach number flows, there is a strong recent interest in the development and analysis of IMEX (implicit/explicit) schemes, which rely on a splitting of the convective flux into stiff and nonstiff parts. A key ingredient of the analysis is the so-called Asymptotic Preserving (AP) property, which guarantees uniform consistency and stability as the Mach number goes to zero. While many authors have focussed on asymptotic consistency, we study asymptotic stability in this paper: does an IMEX scheme allow for a CFL number which is independent of the Mach number? We derive a stability criterion for a general linear hyperbolic system. In the decisive eigenvalue analysis, the advective term, the upwind diffusion and a quadratic term stemming from the truncation in time all interact in a subtle way. As an application, we show that a new class of splittings based on characteristic decomposition, for which the commutator vanishes, avoids the deterioration of the time step which has sometimes been observed in the literature.