Skip to main content
SpringerLink
Log in

On Aposteriori Error Analysis of DG Schemes Approximating Hyperbolic Conservation Laws

  • Conference paper
  • First Online:
Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 77))

  • 1236 Accesses

Abstract

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the \(L^1\) contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in \(L^2,\) based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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. Baccouch, M., Adjerid, S.: Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes. Comput. Methods Appl. Mech. Eng. 200(1–4), 162–177 (2011)

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  4. Dedner, A., Makridakis, C., Ohlberger, M.: Error control for a class of Runge-Kutta discontinuous Galerkin methods for nonlinear conservation laws. SIAM J. Numer. Anal. 45(2), 514–538 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Giesselmann, J., Makridakis, C., Pryer, T.: Aposteriori analysis of discontinuous Galkerin schemes for systems of hyperbolic conservation laws. In preparation

    Google Scholar 

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

    Google Scholar 

  8. Jovanović, V., Rohde, C.: 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  MATH  Google Scholar 

  9. Kröner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69(229), 25–39 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. Makridakis, C., Nochetto, R.H.: A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104(4), 489–514 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ohlberger, M.: A review of a posteriori error control and adaptivity for approximations of non-linear conservation laws. Internat. J. Numer. Methods Fluids 59(3), 333–354 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Jan Giesselmann

  2. Department of Mathematics and Statistics, Whiteknights, University of Reading, Reading, PO Box 220, GB-RG6 6AX, UK

    Tristan Pryer

Authors
  1. Jan Giesselmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tristan Pryer
    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. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Giesselmann, J., Pryer, T. (2014). On Aposteriori Error Analysis of DG Schemes Approximating Hyperbolic Conservation Laws. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_30

Download citation

Publish with us

Policies and ethics

Search

Navigation