Direct Enforcement of Entropy Balance in a Discontinuous Galerkin method
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In order to ensure numerical entropy stability, which for fluid dynamics equations means to respect the second law of thermodynamics, one must guarantee entropy is not destroyed over time. In this regard, Discontinuous Galerkin (DG) methods proved to be well suited to construct entropy conserving/stable schemes. Hughes et al. were among the first to develop an entropy stable formulation by using entropy variables, albeit in the DG framework it requires over-integration with a degradation of the computational performance. A possible approach to avoid over-integration was suggested by Chen and Shu, inspired by the Direct Enforcement of Entropy Balance (DEEB) proposed by Abgrall in the context of residual distribution methods. In our contribution we implement the DEEB in a modal DG scheme, which makes use of entropy variables, with the aim to obtain an entropy conserving/stable solver not requiring over-integration. Entropy variables are here used both directly, as unknowns, and through the entropy projection of conservative variables. The theoretical order of convergence is numerically verified and the robustness of the method assessed on a suite of test cases, including the Richtmeyer-Meshkov and Kelvin-Helmhotz instabilities, showing that DEEB is able to improve robustness significantly while maintaining computational effort almost unaffected.
