A Low Mach Number Solver: Enhancing Applicability

TL;DR

This paper extends a low Mach number solver to Navier-Stokes equations, demonstrating its stability, efficiency, and scalability for very low Mach number flows in astrophysics and meteorology, overcoming limitations of explicit methods.

Contribution

It introduces an extension of Kwatra et al.'s method to more complex equations and stabilizes it for convection problems, highlighting its effectiveness at very low Mach numbers.

Findings

The extended method remains stable down to Mach number 0.001.

It scales efficiently over three orders of magnitude of processor cores.

The method outperforms explicit schemes in low Mach number regimes.

Abstract

In astrophysics and meteorology there exist numerous situations where flows exhibit small velocities compared to the sound speed. To overcome the stringent timestep restrictions posed by the predominantly used explicit methods for integration in time the Euler (or Navier-Stokes) equations are usually replaced by modified versions. In astrophysics this is nearly exclusively the anelastic approximation. Kwatra et al. have proposed a method with favourable time-step properties integrating the original equations (and thus allowing, for example, also the treatment of shocks). We describe the extension of the method to the Navier-Stokes and two-component equations. - However, when applying the extended method to problems in convection and double diffusive convection (semiconvection) we ran into numerical difficulties. We describe our procedure for stabilizing the method. We also investigate the behaviour of Kwatra et al.'s method for very low Mach numbers (down to Ma = 0.001) and point out its very favourable properties in this realm for situations where the explicit counterpart of this method returns absolutely unusable results. Furthermore, we show that the method strongly scales over 3 orders of magnitude of processor cores and is limited only by the specific network structure of the high performance computer we use.