A Conservative Time-Accurate Local Time-Stepping DG Scheme Based on a Weakly Compressible Model for Unsteady Low-Mach-Number Flows

Abstract

This paper presents a conservative high-order discontinuous Galerkin (DG) method featuring time-accurate local time stepping for simulating low-Mach-number unsteady flows, based on a weakly compressible formulation. In this model, pressure is defined solely as a function of density, eliminating the need for a global pressure Poisson equation typical of incompressible solvers while preserving the locality and conservation of compressible schemes. This makes it suitable for low-speed unsteady flows and aeroacoustics. The spatial discretization uses a strong-form nodal DG spectral element method (DGSEM) on Gauss-Lobatto-Legendre points. Inviscid fluxes are handled by numerical fluxes tailored to the weakly compressible system; specifically, a two-rarefaction approximate Riemann solver is developed for the constant-sound-speed barotropic equation of state. Viscous terms employ the incomplete interior penalty Galerkin (IIPG) method. For time integration, a continuous extension Runge-Kutta (CERK) scheme constructs cell-local predictor polynomials for continuous-in-time volume reconstructions. Face fluxes are split into interior and common contributions: the former matches the volume quadrature, while the latter uses piecewise Gaussian quadrature from continuous predictors. This split preserves discrete summation-by-parts cancellation and ensures conservative inter-element flux exchange.