A finite volume scheme for the Euler system inspired by the two velocities approach
This work provides a numerical method with theoretical guarantees for gas dynamics simulations, addressing stability and convergence issues.
The paper proposes a new finite volume scheme for the Euler system that uses upwinding on velocity rather than convected quantities, ensuring positivity of density and pressure, the minimal entropy principle, and convergence to smooth solutions.
We propose a new finite volume scheme for the Euler system of gas dynamics motivated by the model proposed by H. Brenner. Numerical viscosity imposed through upwinding acts on the velocity field rather than on the convected quantities. The resulting numerical method enjoys the crucial properties of the Euler system, in particular positivity of the approximate density and pressure and the minimal entropy principle. In addition, the approximate solutions generate a dissipative measure-valued solutions of the limit system. In particular, the numerical solutions converge to the smooth solution of the system as long as the latter exists.