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

  • Eduard Feireisl
  • Mária Lukáčová-Medvid’ováEmail author
  • Hana Mizerová


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.


Compressible Euler equations Entropy stable finite volume scheme Entropy stability Convergence Dissipative measure-valued solution 

Mathematics Subject Classification

65M08 76N10 35L65 35R06 



  1. 1.
    E. Audussse, F. Bouchut, M.-O. Bristeau, and J. Sainte-Marie. Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system. Math. Comp. 85 (2016), 2815–2837.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    J. J. Alibert, and G. Bouchitté. Non-uniform integrability and generalized Young measures. J. Convex Anal. 4(1) (1997), 129–147.MathSciNetzbMATHGoogle Scholar
  3. 3.
    J.M. Ball. A version of the fundamental theorem for Young measures. In Lect. Notes in Physics 344, Springer-Verlag, 1989, pp. 207–215.Google Scholar
  4. 4.
    Y. Brenier, C. De Lellis, and L. Székelihidi, Jr.. Weak-strong uniqueness for measure-valued solutions Comm. Math. Phys. 305(2) (2011), 351–361.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A. Bressan, G. Crasta, and B. Piccoli. Well-posedness of the Cauchy problem for $n \times n$ systems of conservation laws. Memoirs of the AMS 146(694) (2000).Google Scholar
  6. 6.
    A. Bressan. Uniqueness and stability for one dimensional hyperbolic systems of conservation laws. In XIIIth International Congress on Mathematical Physics (London, 2000), Int. Press, Boston, MA, 2001, pp. 311-317.Google Scholar
  7. 7.
    F. Berthelin, and F. Bouchut. Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy. Meth. Appl. Anal. 9 (2002), 313–327.MathSciNetzbMATHGoogle Scholar
  8. 8.
    F. Bouchut. Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 (2003), 623–672.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    F. Bouchut, and X. Lébrard. Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography. Preprint
  10. 10.
    F. Berthelin. Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws. Numer. Math. 99 (2005), 585–604.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Y. Brenier, C. De Lellis, and L. Székelyhidi, Jr.. Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys. 305(2) (2011), 351–361.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Březina, and E. Feireisl. Measure-valued solutions to the complete Euler system. J. Math. Soc. Jpn. 70(4) (2018), 1227–1245.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. Březina, and E. Feireisl. Maximal dissipation principle for the complete Euler system. Preprint arXiv:1712.04761, 2018.Google Scholar
  14. 14.
    E. Chiodaroli, C. De Lellis, and O. Kreml. Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. 68(7) (2015), 1157–1190.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    C. Christoforou, M. Galanopoulou, and A.E. Tzavaras. A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness. Commun. Part. Diff. Eq. 43(7) (2018), 1019–1050.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    F. Coquel, and P. LeFloch. An entropy satisfying MUSCL scheme for systems of conservation laws. Numer. Math. 74 (1996), 1–33.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    C. M. Dafermos. The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 94 (1979), 373–389.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, New York, 2000.zbMATHCrossRefGoogle Scholar
  19. 19.
    C. De Lellis, and L. Székelyhidi, Jr.. The Euler equations as a differential inclusion. Ann. of Math. 170(2) (2009), 1417–1436.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    C. De Lellis, and L. Székelyhidi, Jr.. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1) (2010), 225–260.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    S. Demoulini, D. M. A. Stuart, and A. E. Tzavaras. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal. 205(3) (2012), 927–961.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    R. DiPerna. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28 (1979), 137–188.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    R. DiPerna. Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983), 27–70.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    R. DiPerna. Measure valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3) (1985), 223–270.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    R. DiPerna, and A. Majda. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4) (1987), 667–689.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda, and E. Wiedemann. Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Partial Differential Equations 55(6) (2016), 55–141.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    E. Feireisl, and M. Lukáčová-Medvid’ová. Convergence of a mixed finite element finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions, Found. Comput. Math. 18(3) (2018), 703–730.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    E. Feireisl, C. Klingenberg, O. Kreml, and S. Markfelder. On oscillatory solutions to the complete Euler system. Preprint arXiv:1710.10918, 2017.Google Scholar
  29. 29.
    M. Feistauer. Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics Series 67, Longman Scientific & Technical, Harlow, 1993.Google Scholar
  30. 30.
    M. Feistauer, J. Felcman, and I. Straškraba. Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford, 2003.zbMATHGoogle Scholar
  31. 31.
    U. Fjordholm. High-order accurate entropy stable numerical schemes for hyperbolic conservation laws. ETH Zürich dissertation Nr. 21025, 2013.Google Scholar
  32. 32.
    U. Fjordholm, S. Mishra, and E. Tadmor. On the computation of measure-valued solutions. Acta Numer. 25 (2016), 567–679.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    U. Fjordholm, S. Mishra, and E. Tadmor. Arbitrary order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws. SIAM J. Num. Anal. 50(2) (2012), 544–573.zbMATHCrossRefGoogle Scholar
  34. 34.
    U. Fjordholm, R. Käppeli, S. Mishra, and E. Tadmor. Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws. Found. Comput. Math. 17 (2017), 763–827.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697–715.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    E. Godlewski, and P.-A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York, 1996.zbMATHCrossRefGoogle Scholar
  37. 37.
    P. Gwiazda, A. Świerczewska-Gwiazda, and E. Wiedemann. Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity 28(11) (2015), 3873–3890.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A. Harten. On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49 (1983), 151–164.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    V. Jovanović, and Ch. Rohde. Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws. SIAM J. Numer. Anal. 43(6) (2006), 2423–2449.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    S. N. Kruzkhov. First order quasilinear equations in several independent variables. USSR Math. Sbornik 10(2) (1970), 217–243.CrossRefGoogle Scholar
  41. 41.
    D. Kröner. Numerical Schemes for Conservation Laws. John Wiley, Chichester, 1997.zbMATHGoogle Scholar
  42. 42.
    D. Kröner,and W. M. Zajaczkowski. Measure-valued solutions of the Euler equations for ideal compressible polytropic fluids. Math. Methods Appl. Sci. 19(3) (1996), 235–252.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    P. LeFloch, J.M. Mercier, and C. Rohde. Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40 (2002), 1968–1992.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Texts in Applied Mathematics, 2002.Google Scholar
  45. 45.
    P.-L. Lions. Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, 1996.Google Scholar
  46. 46.
    A. Mielke. Flow properties for Young-measure solutions of semilinear hyperbolic problems. Proc. R. Soc. Edin. A-MA 129(1) (1999), 85–123.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    P. Pedregal. Parametrized Measures and Variational Principles. Birkhäuser, Basel, 1997.zbMATHCrossRefGoogle Scholar
  48. 48.
    B. Perthame, and C.-W. Shu. On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73 (1996), 119–130.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    S. Schochet. Examples of measure-valued solutions. Commun. Part. Diff. Eq. 14(5) (1989), 545–575.MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    D. Serre. Systems of Conservation Laws, 1: Hyperbolicity, Entropies, Shock Waves (English translation). Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
  51. 51.
    L. Székelyhidi, Jr., and E. Wiedemann. Young measures generated by ideal incompressible fluid flows. Arch. Rational Mech. Anal. 206 (2012), 333–366.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    E. Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comp. 49(179) (1987), 91–103.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    E. Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems. Acta Numer. 12 (2003), 451–512.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    E. Tadmor. Minimum entropy principle in the gas dynamic equations Appl. Num. Math. 2 (1986), 211–219.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    E. Wiedemann. Weak-strong uniqueness in fluid dynamics. Partial differential equations in fluid mechanics, 289–326, London Math. Soc. Lecture Note Ser. 452, Cambridge Univ. Press, 2018.Google Scholar

Copyright information

© SFoCM 2019

Authors and Affiliations

  • Eduard Feireisl
    • 1
  • Mária Lukáčová-Medvid’ová
    • 2
    Email author
  • Hana Mizerová
    • 1
  1. 1.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic
  2. 2.Institute of MathematicsJohannes Gutenberg-University MainzMainzGermany

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