Flux Splitting for stiff equations: A notion on stability
For computational fluid dynamics researchers, this provides a theoretical guarantee of asymptotic stability for IMEX schemes, addressing a known bottleneck in low Mach number simulations.
The paper derives a stability criterion for IMEX schemes applied to low Mach number flows, showing that a new class of splittings based on characteristic decomposition avoids time step deterioration, achieving a CFL number independent of Mach number.
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.