On the Relaxation Technique Applied to Linearly Implicit Rosenbrock Schemes for a Fully-Discrete Entropy Conserving/Stable dG Method

In this work, a high-order modal discontinuous Galerkin (dG) method is employed to solve the Euler equations using entropy variables. Entropy conservation and stability are ensured at the spatial semi-discrete level through entropy-conserving/stable numerical fluxes and the over-integration technique. For time integration, linearly implicit Rosenbrock-type Runge–Kutta schemes are used. However, since these schemes are not provably entropy-conserving/stable, their use to predict unsteady flows may lead to solutions that lack the desired entropy properties. To address this issue, a relaxation technique is applied to enforce entropy conservation or stability at the fully discrete level. The accuracy, conservation/stability properties and robustness of the fully-discrete scheme equipped with the relaxation technique are assessed through the following numerical experiments: (1) the isentropic vortex, (2) the Kelvin-Helmholtz instability, (3) the Taylor–Green vortex.