Skip to main content
SpringerLink
Log in

Residual Error Indicators for Discontinuous Galerkin Schemes for Discontinuous Solutions to Systems of Conservation Laws

  • Conference paper
  • First Online:
Theory, Numerics and Applications of Hyperbolic Problems I (HYP 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 236))

  • 1185 Accesses

Abstract

This contribution is devoted to fully discrete discontinuous Galerkin approximations of systems of hyperbolic conservation laws in one space dimension. Its focus is on a posteriori error estimators which are obtained by a combination of a reconstruction approach with the relative entropy stability framework. It was shown in earlier works that for certain numerical fluxes, the error estimators are of the same order as the true error before shock formation. For discontinuous solutions, the use of the relative entropy methodology prevents convergence of the error estimator. We investigate whether a part of the error estimator (related to residuals) is convergent post-shock and whether it is useful as an error indicator or a smoothness indicator.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C. Arvanitis, C. Makridakis, A.E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws. SIAM J. Numer. Anal. 42(4), 1357–1393 (2004)

    Article  MathSciNet  Google Scholar 

  2. E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible euler system. J. Hyperbolic Differ. Equ. 11(03), 493–519 (2014)

    Article  MathSciNet  Google Scholar 

  3. C.M. Dafermos, The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70(2), 167–179 (1979)

    Article  MathSciNet  Google Scholar 

  4. C.M. Dafermos, Hyperbolic conservation laws in continuum physics, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 3rd edn (Springer, Berlin, 2010)

    Google Scholar 

  5. C. De Lellis, L. Székelyhidi, On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)

    Article  MathSciNet  Google Scholar 

  6. A. Dedner, J. Giesselmann, A posteriori analysis of fully discrete method of lines discontinuous Galerkin schemes for systems of conservation laws. SIAM J. Numer. Anal. 54(6), 3523–3549 (2016)

    Article  MathSciNet  Google Scholar 

  7. A. Dedner, R. Klöfkorn, A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems. J. Sci. Comput. 47(3), 365–388 (2011)

    Article  MathSciNet  Google Scholar 

  8. R.J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28(1), 137–188 (1979)

    Article  MathSciNet  Google Scholar 

  9. J. Giesselmann, C. Makridakis, T. Pryer, A posteriori analysis of discontinuous galerkin schemes for systems of hyperbolic conservation laws. SIAM J. Numer. Anal. 53, 1280–1303 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. R. Hartmann, P. Houston, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24(3), 979–1004 (2002)

    Article  MathSciNet  Google Scholar 

  12. V. Jovanović, C. Rohde, Finite-volume schemes for Friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Numer. Methods Partial Differ. Equ. 21(1), 104–131 (2005)

    Article  MathSciNet  Google Scholar 

  13. V. Jovanović, C. Rohde, Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws. SIAM J. Numer. Anal. 43(6), 2423–2449 (2006)

    Article  MathSciNet  Google Scholar 

  14. H. Kim, M. Laforest, D. Yoon, An adaptive version of Glimm’s scheme. Acta Math. Sci. Ser. B Engl. Ed. 30(2), 428–446 (2010)

    Article  MathSciNet  Google Scholar 

  15. M. Laforest, A posteriori error estimate for front-tracking: systems of conservation laws. SIAM J. Math. Anal. 35(5), 1347–1370 (2004)

    Article  MathSciNet  Google Scholar 

  16. M. Laforest, An a posteriori error estimate for Glimm’s scheme, in Hyperbolic Problems: Theory, Numerics, Applications (Springer, Berlin, 2008), pp. 643–651

    Chapter  Google Scholar 

  17. C. Makridakis, Space and time reconstructions in a posteriori analysis of evolution problems, in ESAIM Proceedings. Vol. 21 (2007) [Journées d’Analyse Fonctionnelle et Numérique en l’honneur de Michel Crouzeix], vol. 21 of ESAIM Proceedings (EDP Sci., Les Ulis, 2007), pp. 31–44

    Google Scholar 

  18. Q. Zhang, C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 44(4), 1703–1720 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

A. Dedner would like to acknowledge support from the Royal Society under its International Exchanges Award.

Author information

Authors and Affiliations

  1. Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK

    Andreas Dedner

  2. University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Jan Giesselmann

Authors
  1. Andreas Dedner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Giesselmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Giesselmann .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dedner, A., Giesselmann, J. (2018). Residual Error Indicators for Discontinuous Galerkin Schemes for Discontinuous Solutions to Systems of Conservation Laws. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_35

Download citation

Publish with us

Policies and ethics

Search

Navigation