Skip to main content
SpringerLink
Log in

Entropy stable schemes for initial-boundary-value conservation laws

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We consider initial-boundary-value problems for systems of conservation laws and design entropy stable finite difference schemes to approximate them. The schemes are shown to be entropy stable for a large class of systems that are equipped with a symmetric splitting, derived from the entropy formulation. Numerical examples for the Euler equations of gas dynamics are presented to illustrate the robust performance of the proposed method.

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

Similar content being viewed by others

References

  1. Barth T.: On discontinuous Galerkin approximations of Boltzmann moment systems with Levermore closure. Comput. Methods Appl. Mech. Eng. 195, 3311–3330 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Du Bois F., Le Floch P.: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71(1), 93–122 (1988)

    Article  MathSciNet  Google Scholar 

  3. Cockburn B., Coquel F., Le Floch P.G.: Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32(3), 687–705 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gerritsen, M.: Designing an Efficient Solution Strategy for Fluid Flows. PhD thesis, Stanford University (1996)

  5. Gerritsen M., Olsson P.: Designing an efficient solution strategy for fluid flows: 1. A stable high order finite difference scheme and sharp shock resolution for the euler equations. J. Comput. Phys. 129(2), 245–262 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gerritsen M., Olsson P.: Designing an efficient solution strategy for fluid flows: II. Stable high-order central finite difference schemes on composite adaptive grids with sharp shock resolution. J. Comput. Phys. 147(2), 293–317 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hou T.Y., Le Floch P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62(206), 497–530 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mock M.S.: Systems of conservation laws of mixed type. J. Differ. Equ. 37, 70–88 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Olsson P.: Summation by parts, projections, and stability I. Math. Comput. 64, 1035 (1995a)

    Article  MathSciNet  MATH  Google Scholar 

  10. Olsson P.: Summation by parts, projections, and stability II. Math. Comput. 64, 1473 (1995b)

    Article  MathSciNet  MATH  Google Scholar 

  11. Olsson, P., Oliger, J.: Energy and Maximum Norm Estimates for Nonlinear Conservation Laws. Technical Report 94.01, The Research Institute of Advanced Computer Science (1994)

  12. Svärd M., Mishra S.: Shock capturing artificial dissipation for high-order finite difference schemes. J. Sci. Comput. 39, 454–484 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shu C.-W., Osher S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, ii. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sod G.: A survey of several finite difference methods for systems of conservation laws. J. Comput. Phys. 17, 1–31 (1978)

    Article  MathSciNet  Google Scholar 

  15. Sonar T.: Entropy production in second-order three-point schemes. Numer. Math. 62, 371–390 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tadmor E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK

    Magnus Svärd

  2. Applied Mathematics, Department of Mathematics, ETH Zürich, Rämistrasse 101, HG G 57.2, 8092, Zürich, Switzerland

    Siddhartha Mishra

Authors
  1. Magnus Svärd
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siddhartha Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magnus Svärd.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Svärd, M., Mishra, S. Entropy stable schemes for initial-boundary-value conservation laws. Z. Angew. Math. Phys. 63, 985–1003 (2012). https://doi.org/10.1007/s00033-012-0216-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-012-0216-x

Mathematics Subject Classification

Keywords

Search

Navigation