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Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

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Abstract

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. 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.

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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, 551 28, Mainz, Germany

    Mária Lukáčová-Medvid’ová

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  1. Eduard Feireisl
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  2. Mária Lukáčová-Medvid’ová
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Correspondence to Mária Lukáčová-Medvid’ová.

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Communicated by Eitan Tadmor.

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The research of E. Feireisl and H. Mizerová leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO: 67985840. The research of M. Lukáčová-Medvid’ová was supported by the German Science Foundation under the Collaborative Research Centers TRR 146 Multiscale Simulation Methods for Soft Matter Systems and TRR 165 Waves to Weather.

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Feireisl, E., Lukáčová-Medvid’ová, M. & Mizerová, H. Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions. Found Comput Math 20, 923–966 (2020). https://doi.org/10.1007/s10208-019-09433-z

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