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

Abstract

We propose a new arbitrary high order accurate semi-implicit space–time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier–Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows.The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier–Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space–time DG scheme for the incompressible Navier–Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In addition, all the volume and surface integrals needed by the scheme depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessing stage. This leads to significant savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator, but on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p=4.