A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

摘要

We propose a new arbitrary high order accurate semi-implicit space-timediscontinuous Galerkin (DG) method for the solution of the two and threedimensional compressible Euler and Navier-Stokes equations on staggeredunstructured curved meshes. The method is pressure-based and semi-implicit andis able to deal with all Mach number flows. In our scheme, the discretepressure is defined on the primal grid, while the discrete velocity field andthe density are defined on a face-based staggered dual grid. All convectiveterms are discretized explicitly, while the pressure terms appearing in themomentum and energy equation are discretized implicitly. Substitution of themomentum equation into the energy equation yields a linear system for thescalar pressure as the only unknown. The enthalpy and the kinetic energy aretaken explicitly and are then updated using a simple Picard procedure. Thanksto the use of a staggered grid, the final pressure system is a very sparseblock five-point system for three dimensional problems. The viscous terms andthe heat flux are also discretized making use of the staggered grid by definingthe viscous stress tensor and the heat flux vector on the dual grid, whichcorresponds to the use of a lifting operator on the dual grid. The time step ofour new numerical method is limited by a CFL condition based only on the fluidvelocity and not on the sound speed. This makes the method particularlyinteresting for low Mach number flows. Finally, a very simple combination ofartificial viscosity and the a posteriori MOOD technique allows to deal withshock waves and thus permits also to simulate high Mach number flows. We showcomputational results for a large set of two and three-dimensional benchmarkproblems, including both low and high Mach number flows and using polynomialapproximation degrees up to p=4.