A stable, efficient and accurate high-order discretization for low Mach flows

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We propose, further develop, and analyze a particularly efficient method to compute singularly perturbed problems such as the compressible Navier-Stokes equations at low Mach number. A key contribution is a new splitting of the flux into fast and slow components, which is based on a suitable reference solution (RS). This is combined with the well-known implicit-explicit (IMEX) paradigm. We call the resulting scheme RS-IMEX. Recent studies of low order finite volume schemes for prototype hyperbolic systems, and high order IMEX Runge-Kutta and IMEX BDF schemes for stiff ODEs have shown increased accuracy.In this project, we generalize the RS-IMEX approach to high order IMEX discontinuous Galerkin methods. In order to build an efficient solver, we need to re-develop several essential ingredients in such a way that they are tailored to the RS-IMEX context: preconditioners based on the reference solution, linear solvers adapted to the extreme stiffness, hybrid IMEX DG discretizations reducing the dimension of the implicit problem, and an efficient coupling of compressible and incompressible solvers, the latter one being particularly important to efficiently employ the reference solution. Throughout the development we analyze the asymptotic consistency and stability of each new component, and compare the resulting methods to recent state of the art IMEX schemes.Our goal is to establish the DG-based RS-IMEX scheme as a general, systematic, particularly accurate, stable and efficient method for asymptotically stiff problems.