Skip to main content
SpringerLink
Log in

Entropic Regularization of the Discontinuous Galerkin Method in Conservative Variables for Two-Dimensional Euler Equations

  • Published:
Mathematical Models and Computer Simulations Aims and scope

Abstract

An entropic regularization of the conservative stable discontinuous Galerkin method (DGM) in conservative variables for two-dimensional Euler equations is constructed based on a special slope limiter. This limiter ensures the fulfillment of two-dimensional analogs of the monotonicity conditions and a discrete analog of the entropic inequality. The developed method was tested on two-dimensional model gas-dynamic problems.

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.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Similar content being viewed by others

REFERENCES

  1. E. Tadmor, “Entropy stable schemes,” Handb. Numer. Anal. 17, 467–493 (2016). https://doi.org/10.1016/bs.hna.2016.09.006

    Article  MathSciNet  Google Scholar 

  2. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Soc. Ind. A-ppl. Math., Philadelphia, 1973). https://doi.org/10.1137/1.9781611970562

  3. S. Osher, “Riemann solvers, the entropy condition, and difference approximations,” SIAM J. Numer. Anal. 21 (2), 217–235 (1984). https://doi.org/10.1137/0721016

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Bouchut, Ch. Bourdarias, and B. Perthame, “A MUSCL method satisfying all the numerical entropy inequalities,” Math. Comput. 65 (216), 1439–1461 (1996). https://doi.org/10.1090/S0025-5718-96-00752-1

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Tadmor, “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems,” Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Ismail and P. L. Roe, “Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks,” J. Comput. Phys. 228 (15), 5410–5436 (2009). https://doi.org/10.1016/j.jcp.2009.04.021

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Chandrashekar, “Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations,” Commun. Comput. Phys. 14 (5), 1252–1286 (2013). https://doi.org/10.4208/cicp.170712.010313a

    Article  MathSciNet  MATH  Google Scholar 

  8. U. S. Fjordholm, S. Mishra, and E. Tadmor, “Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws,” SIAM J. Numer. Anal. 50 (2), 544–573 (2012). https://doi.org/10.1137/110836961

    Article  MathSciNet  MATH  Google Scholar 

  9. X. Cheng and Y. Nie, “A third-order entropy stable scheme for hyperbolic conservation laws,” J. Hyperbolic Differ. Equat. 13 (1), 129–145 (2016). https://doi.org/10.1142/S021989161650003X

    Article  MathSciNet  MATH  Google Scholar 

  10. V. V. Ostapenko, “Symmetric compact schemes with higher order conservative artificial viscosities,” Comput. Math. Math. Phys. 7 (1), 980–999 (2002).

    Google Scholar 

  11. A. A. Zlotnik, “Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations,” Comput. Math. Math. Phys. 57 (4), 706–725 (2017). https://doi.org/10.1134/S0965542517020166

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Cockburn, “An introduction to the Discontinuous Galerkin method for convection-dominated Problems,” in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Ed. by A. Quarteroni, Lecture Notes in Mathematics, Vol. 1697 (Springer, Berlin, 1997), pp. 150–268. https://doi.org/10.1007/BFb0096353.

  13. G. J. Gassner, A. R. Winters, and D. A. Kopriva, “A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations,” Appl. Math. Comput. 272 (Part 2), 291–308 (2016). https://doi.org/10.1016/j.amc.2015.07.014

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Krais, A. Beck, T. Bolemann, H. Frank, D. Flad, G. Gassner, F. Hindenlang, M. Hoffmann, T. Kuhn, M. Sonntag, and C.-D. Munz, “FLEXI: A high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws,” Comput. Math. Appl. 81, 186–219 (2021). https://doi.org/10.1016/j.camwa.2020.05.004

    Article  MathSciNet  MATH  Google Scholar 

  15. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Application of averaging to smooth the solution in the DG method,” KIAM Preprint No. 89 (Keldysh Inst. Appl. Math. RAS, Moscow, 2017) [in Russian]. https://doi.org/10.20948/prepr-2017-89.

  16. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Impact of different limiting functions on the order of solution obtained by RKDG,” Math. Models Comput. Simul. 5 (4), 346–349 (2013). https://doi.org/10.1134/S2070048213040091

    Article  MathSciNet  MATH  Google Scholar 

  17. M. E. Ladonkina and V. F. Tishkin, “Godunov method: a generalization using piecewise polynomial approximations,” Differ. Equat. 51 (7), 895–903 (2015). https://doi.org/10.1134/S0012266115070083

    Article  MathSciNet  MATH  Google Scholar 

  18. M. E. Ladonkina and V. F. Tishkin, “On Godunov-type methods of high order of accuracy,” Dokl. Math. 91 (2), 189–192 (2015). https://doi.org/10.1134/S1064562415020222

    Article  MathSciNet  MATH  Google Scholar 

  19. V. F. Tishkin, V. T. Zhukov, and E. E. Myshetskaya, “Justification of Godunov’s scheme in the multidimensional case,” Models Comput. Simul. 8 (5), 548–556 (2016). https://doi.org/10.1134/S2070048216050124

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu. A. Kriksin and V. F. Tishkin, “Entropic regularization of Discontinuous Galerkin method in one-dimensional problems of gas dynamics,” KIAM Preprint No. 100 (Keldysh Inst. Appl. Math. RAS, Moscow, 2018) [in Russian]. https://doi.org/10.20948/prepr-2018-100.

  21. M. D. Bragin, Yu. A. Kriksin, and V. F. Tishkin, “Verification of an entropic regularization method for discontinuous Galerkin schemes applied to hyperbolic equations,” KIAM Preprint No. 18 (Keldysh Inst. Appl. Math. RAS, Moscow, 2019) [in Russian]. https://doi.org/10.20948/prepr-2019-18.

  22. M. D. Bragin, Y. A. Kriksin, and V. F. Tishkin, “Discontinuous Galerkin method with an entropic slope limiter for Euler equations,” Math. Models Comput. Simul. 12 (5), 824–833 (2020). https://doi.org/10.1134/S2070048220050038

    Article  MathSciNet  MATH  Google Scholar 

  23. Yu. A. Kriksin and V. F. Tishkin, “Numerical solution of the Einfeldt problem based on the discontinuous Galerkin method,” KIAM Preprint No. 90 (Keldysh Inst. Appl. Math. RAS, Moscow, 2019) [in Russian]. https://doi.org/10.20948/prepr-2019-90.

  24. Yu. A. Kriksin and V. F. Tishkin, “Entropy-stable discontinuous Galerkin method for Euler equations using nonconservative variables,” Math. Models Comput. Simul. 13 (3), 416–425 (2021). https://doi.org/10.1134/S2070048221030091

    Article  MathSciNet  Google Scholar 

  25. M. D. Bragin, Y. A. Kriksin, and V. F. Tishkin, “Entropy-stable discontinuous Galerkin method for two-dimensional Euler equations,” Math. Models Comput. Simul. 13 (5), 897–906 (2021). https://doi.org/10.1134/S2070048221050069

    Article  MathSciNet  MATH  Google Scholar 

  26. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, A. N. Kraiko, and G. P. Prokopov, Numerical Solution of Multidimensional Problems of Gas Dynamics (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  27. R. Liska and B. Wendroff, “Comparison of several difference schemes on 1D and 2D test problems for the Euler equations,” SIAM J. Sci. Comput. 25 (3), 995–1017 (2003). https://doi.org/10.1137/S1064827502402120

    Article  MathSciNet  MATH  Google Scholar 

  28. M. D. Bragin and B.V. Rogov, “Conservative limiting method for high-order bicompact schemes as applied to systems of hyperbolic equations,” Appl. Numer. Math. 151, 229–245 (2020). https://doi.org/10.1016/j.apnum.2020.01.005

    Article  MathSciNet  MATH  Google Scholar 

  29. M. D. Bragin, “Entropy stability of bicompact schemes in gas dynamics problems,” Math. Models Comput. Simul. 13 (4), 613–622 (2021). https://doi.org/10.1134/S2070048221040086

    Article  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors thank the Supercomputer Center of the Research Computing Center of Moscow State University and the Center for Information Technology of the University of Groningen (Netherlands) for the possibility of performing the calculations.

Funding

This study was supported by a grant from the Russian Science Foundation (project no. 21-11-00198).

Author information

Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    M. D. Bragin, Y. A. Kriksin & V. F. Tishkin

Authors
  1. M. D. Bragin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. A. Kriksin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. F. Tishkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. D. Bragin, Y. A. Kriksin or V. F. Tishkin.

Ethics declarations

The authors declare that they have no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bragin, M.D., Kriksin, Y.A. & Tishkin, V.F. Entropic Regularization of the Discontinuous Galerkin Method in Conservative Variables for Two-Dimensional Euler Equations. Math Models Comput Simul 14, 578–589 (2022). https://doi.org/10.1134/S2070048222040056

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070048222040056

Keywords:

Search

Navigation