Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions

0531438 - MÚ 2021 RIV US eng J - Journal Article
Feireisl, Eduard - Lukáčová-Medviďová, M. - Mizerová, Hana
Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions.
Foundations of Computational Mathematics. Roč. 20, č. 4 (2020), s. 923-966. ISSN 1615-3375. E-ISSN 1615-3383
EU Projects: European Commission(XE) 320078 - MATHEF
Institutional support: RVO:67985840
Keywords : compressible Euler equations * convergence * dissipative measure-valued solution * entropy stability * entropy stable finite volume scheme
OECD category: Pure mathematics
Impact factor: 2.987, year: 2020
Method of publishing: Limited access
https://doi.org/10.1007/s10208-019-09433-z

The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. 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. Here dissipative means that a suitable form of the second law of thermodynamics is incorporated in the definition of the measure-valued solutions.
Permanent Link: http://hdl.handle.net/11104/0310103