| Issue |
ESAIM: M2AN
Volume 58, Number 1, January-February 2024
|
|
|---|---|---|
| Page(s) | 47 - 77 | |
| DOI | https://doi.org/10.1051/m2an/2023093 | |
| Published online | 16 January 2024 | |
A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system
School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Thiruvananthapuram 695551, India
* Corresponding author: amoghk0720@iisertvm.ac.in
Received:
19
June
2023
Accepted:
10
November
2023
A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [Feireisl et al., Calc. Var. Part. Differ. Equ. 55 (2016) 141] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax’s equivalence theorem under some physically reasonable boundedness assumptions. The concept of Ƙ-convergence [Feireisl et al., IMA J. Numer. Anal. 40 (2020) 2227–2255] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak–strong uniqueness principle and relative entropy are presented.
Mathematics Subject Classification: 35L45 / 35L60 / 35L65 / 35L67 / 35D99 / 35R06 / 65M08
Key words: Compressible Euler system / dissipative measure-valued solution / Young measure / finite volume method / entropy stability / consistency / Ƙ-convergence
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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