Entropy-Stable Discontinuous Galerkin Method for Euler Equations Using Nonconservative Variables

A conservative version of the entropy stable discontinuous Galerkin method is proposed for the Euler equations in variables: density, momentum density, and pressure. For the equation describing the dynamics of the mean pressure in an FE, a special difference approximation in time and conservative in total energy is constructed. The entropy inequality and the requirements for the monotonicity of the numerical solution are satisfied by a special slope limiter. The developed method is successfully tested on a number of model gas-dynamic problems. In particular, in the numerical solution of the Einfeldt problem, the quality of the calculation of the specific internal energy is significantly improved.