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A finite volume scheme for the Euler system inspired by the two velocities approach

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Abstract

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.

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Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Eduard Feireisl & Hana Mizerová

  2. Institute of Mathematics, Johannes Gutenberg-University Mainz, Staudingerweg 9, 55128, Mainz, Germany

    Mária Lukáčová-Medvid’ová

  3. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava, Slovakia

    Hana Mizerová

Authors
  1. Eduard Feireisl
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  2. Mária Lukáčová-Medvid’ová
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  3. Hana Mizerová
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Correspondence to Mária Lukáčová-Medvid’ová.

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The research of EF and HM leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840. The research of ML was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the Transregional Collaborative Research Centers SFB/TRR 146 (Project number 233630050) and SFB/TRR 165 Waves to Weather (Project A2).

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Feireisl, E., Lukáčová-Medvid’ová, M. & Mizerová, H. A finite volume scheme for the Euler system inspired by the two velocities approach. Numer. Math. 144, 89–132 (2020). https://doi.org/10.1007/s00211-019-01078-y

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  • DOI: https://doi.org/10.1007/s00211-019-01078-y

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