Skip to main content
SpringerLink
Log in

From Suitable Weak Solutions to Entropy Viscosity

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper focuses on the notion of suitable weak solutions for the three-dimensional incompressible Navier-Stokes equations and discusses the relevance of this notion to Computational Fluid Dynamics. The purpose of the paper is twofold (i) to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they might relate to LES (ii) to introduce an entropy viscosity technique based on the notion of suitable weak solution and to illustrate numerically this concept.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adams, N.A., Stolz, S.: A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys. 178(2), 391–426 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardos, C., le Roux, A.Y., Nédélec, J.-C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)

    Article  MATH  Google Scholar 

  3. Bertoluzza, S.: The discrete commutator property of approximation spaces. C. R. Acad. Sci. Paris, Sér. I 329(12), 1097–1102 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35(6), 771–831 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Emery, A.F.: An evaluation of several differencing methods for inviscid fluid flow problems. J. Comput. Phys. 2, 306–331 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guermond, J.-L.: Finite-element-based Faedo-Galerkin weak solutions to the Navier-Stokes equations in the three-dimensional torus are suitable. J. Math. Pures Appl. 85(3), 451–464 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guermond, J.-L.: Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl. 88, 87–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guermond, J.-L.: On the use of the notion of suitable weak solutions in CFD. Int. J. Numer. Methods Fluids 57, 1153–1170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guermond, J.-L., Pasquetti, R.: Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C. R. Math. Acad. Sci. Paris 346, 801–806 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Guermond, J.-L., Pasquetti, R.: Entropy viscosity method for high-order approximations of conservation laws. In: Ronquist, E. (ed.) ICOSAHOM09. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 411–418. Springer, Berlin (2011)

    Google Scholar 

  12. Guermond, J.-L., Prudhomme, S.: On the construction of suitable solutions to the Navier-Stokes equations and questions regarding the definition of large eddy simulation. Physica D 207, 64–78 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoffman, J., Johnson, C.: Computational Turbulent Incompressible Flow. Applied Mathematics: Body and Soul, vol. 4. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  14. Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    MathSciNet  MATH  Google Scholar 

  15. Johnson, C., Szepessy, A.: On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comput. 49(180), 427–444 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Johnson, C., Szepessy, A., Hansbo, P.: On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comput. 54(189), 107–129 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N. S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  18. Kurganov, A., Petrova, G., Popov, B.: Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29(6), 2381–2401 (2007) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  20. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  21. Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Commun. Pure Appl. Math. 51(3), 241–257 (1998)

    Article  MATH  Google Scholar 

  22. Liska, R., Wendroff, B.: Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25(3), 995–1017 (2003) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  23. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55(2), 97–112 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Jean-Luc Guermond & Bojan Popov

  2. Lab. J.A. Dieudonné, UMR CNRS 6621, Université de Nice-Sophia Antipolis, Nice, France

    Richard Pasquetti

Authors
  1. Jean-Luc Guermond
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Pasquetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bojan Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Luc Guermond.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guermond, JL., Pasquetti, R. & Popov, B. From Suitable Weak Solutions to Entropy Viscosity. J Sci Comput 49, 35–50 (2011). https://doi.org/10.1007/s10915-010-9445-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9445-3

Keywords

Search

Navigation