Numerical solution of the Einfeldt problem based on the discontinuous Galerkin method

The numerical algorithm for solving Euler equations based on the new variational principle of deriving the modified equations of the discontinuous Galerkin method is developed. As the sought variables which depend on time and space the gas density, momentum density and pressure are used. The corresponding numerical solutions satisfy discrete analogues of the conservation laws of mass, momentum, total energy, and entropic inequality. The Einfeldt problem is considered as an example illustrating the effectiveness of the developed algorithm. Numerical calculations show a significant improvement in the quality of the resulting approximate solutions.