A Comparative Study of Different Sets of Variables in a Discontinuous Galerkin Method with Entropy Balance Enforcement

This paper investigates the effectiveness of different sets of variables in solving the compressible Euler equations using a modal Discontinuous Galerkin framework. Alongside the commonly used conservative and primitive variables, the entropy and logarithmic sets are considered to enforce entropy conservation/stability and positivity preservation of the thermodynamic state, respectively. An explicit correction to enforce entropy conservation/stability at the discrete level is also considered, with a significant increase in robustness for some of the solution strategies. Several two-dimensional inviscid test cases are computed to compare the performance of the different sets of variables, adding a directional shock-capturing term to the discretised equations when necessary. The entropy and logarithmic sets proved to be the most robust, completing simulations of an astrophysical jet at Mach number 2000 up to polynomial degree seven.