On the performance of different sets of variables for the discontinuous Galerkin discretization of the Euler equations
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The governing equations of fluid dynamics are usually formulated in terms of conservative or primitive variables. An alternative formulation has been developed to guarantee entropy conservation, leading to the definition of the set of entropy variables. In the literature, the numerical investigation of the different approaches, which are identical only at continuous level, is mainly limited to low-order schemes, such as standard Finite Volumes. Recently, the use of the entropy variables in a high-order framework, such as Discontinuous Galerkin (DG) discretization, has drawn attention. In this work, we investigate the effect of solving for different sets of variables in a modal DG framework. In particular, we consider the implementation of the following alternatives: conservative variables; entropy variables; primitive variables based on pressure and temperature: primitive variable based on the logarithms of pressure and temperature to ensure the positivity of all the thermodynamic quantities at the discrete level. An assessment of the performance of the different choices of variables is provided by presenting several canonical test-cases and focusing on the robustness and conservation properties of each discretization.
