Abstract

We present the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We study the classical schemes based on the Lax-Friedrichs numerical flux [1], and moreover propose a new finite volume scheme motivated by the Brenner model [2]. Numerical viscosity imposed through upwinding in the latter scheme acts on the velocity field rather than on the convected quantities. For both methods we establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter. Finally, we discuss convergence of a finite volume scheme for the compressible (barotropic) Navier-Stokes equations without the a priori hypothesis of the existence of smooth solution [3]. Numerical experiments for benchmark tests support our theoretical results.

These results are based on a joint work with Eduard Feireisl (Czech Academy of Sciences in Prague), Mária Lukáčová-Medvid’ová (Johannes Gutenberg University in Mainz) and Bangwei She (Czech Academy of Sciences in Prague).

References

[1] E. Feireisl, M. Mária Lukáčová-Medvid’ová, and H. Mizerová. Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions.
arXiv:1803.08401, 2018.

[2] E. Feireisl, M. Mária Lukáčová-Medvid’ová, and H. Mizerová. A finite volume scheme for the Euler system inspired by the two velocities approach.
arXiv:1805.05072, 2018.

[3] E. Feireisl, M. Mária Lukáčová-Medvid’ová, H. Mizerová, and B. She. Convergence of a finite volume scheme for the compressible Navier-Stokes system.
arXiv:1811.02866, 2018.