Advertisement

Exponential Scaling and the Time Growth of the Error of DG for Advection-Reaction Problems

  • Václav KučeraEmail author
  • Chi-Wang Shu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We present an overview of the results of the authors’ paper (Kučera and Shu, IMA J Numer Anal, to appear) on the time growth of the error of the discontinuous Galerkin (DG) method and set them in appropriate context. The application of Gronwall’s lemma gives estimates which grow exponentially in time even for problems where such behavior does not occur. In the case of a nonstationary advection-diffusion equation we can circumvent this problem by considering a general space-time exponential scaling argument. Thus we obtain error estimates for DG which grow exponentially not in time, but in the time particles carried by the flow field spend in the spatial domain. If this is uniformly bounded, one obtains an error estimate of the form C(hp+1∕2), where C is independent of time. We discuss the time growth of the exact solution and the exponential scaling argument and give an overview of results from Kučera and Shu (IMA J Numer Anal, to appear) and the tools necessary for the analysis.

Notes

Acknowledgements

The work of V. Kučera was supported by the J. William Fulbright Commission in the Czech Republic and research project No. 17-01747S of the Czech Science Foundation. The work of C.-W. Shu was supported by DOE grant DE-FG02-08ER25863 and NSF grants DMS-1418750 and DMS-1719410.

References

  1. 1.
    B. Ayuso, L.D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47(2), 1391–1420 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Devinatz, R. Ellis, A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, II. Indiana Univ. Math. J. 23, 991–1011 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Feistauer, K. Švadlenka, Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12(2), 97–117 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46(173), 1–26 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems. IMA J. Numer. Anal. 34(2), 820–861 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Kučera, C.-W. Shu, On the time growth of the error of the DG method for advective problems. IMA J. Numer. Anal. https://doi.org/10.1093/imanum/dry013Google Scholar
  7. 7.
    U. Nävert, A finite element method for convection-diffusion problems, Ph.D. Thesis, Chalmers University of Technology, 1982Google Scholar
  8. 8.
    W.H. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73–479, 1973Google Scholar
  9. 9.
    Q. Zhang, C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42(2), 641–666 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations