A low Mach number solver: Enhancing applicability

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. (2009) [19] have proposed a method with favorable 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 behavior of Kwatra et al.’s method for very low Mach numbers (down to Ma = 0.001) and point out its very favorable 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 three orders of magnitude of processor cores and is limited only by the specific network structure of the high performance computer we use.