On the entropy conserving/stable implicit DG discretization of the Euler equations in entropy variables

The aim of this paper is to investigate the behavior of a high-order accurate Discontinuous Galerkin entropy conserving/stable scheme in space for unsteady compressible inviscid flows. In order to ensure an entropy conserving/stable scheme in space, several entropy conserving/stable numerical fluxes are considered. For the time discretization, high-order accurate linearly implicit Rosenbrock-type Runge–Kutta schemes are used. These schemes cannot be provably entropy conserving/stable, but, for an enough small time step size, the error due to the time integration has negligible contribution, and entropy conserving/stable properties can be fulfilled. The properties of the fully discrete system of equations are assessed in a series of numerical experiments of growing complexity, i.e.: (i) the isentropic vortex convection problem, (ii) the double shear layer, (iii) the Sod shock tube and (iv) the Taylor–Green vortex. For each one of these test-case the accuracy and the related order of convergence of the time integration schemes and of the numerical fluxes employed will be investigated, as well as the comparison between the accuracy provided by the entropy and the primitive set of variables and their stability properties. Furthermore, same relevant issues related to the use of entropy conserving fluxes are addressed, like the necessity to use higher accurate quadrature rules, the spatial sub-optimal order of convergence, and the behavior of the fully discrete system of equations when long-time simulations are performed.