Skip to main content
SpringerLink
Log in

The Numerical Study of Singular Shocks Regularized by Small Viscosity

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A singular shock is a measure valued solution found by considering viscosity limit solutions to certain hyperbolic systems. In this work, some fundamental properties concerning such solutions are derived and then illustrated by extensive numerical study.

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. Dafermos, C. M. (2000). Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin-Heidelberg.

    Google Scholar 

  2. Glimm, J. (1965). Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 95-105.

    Google Scholar 

  3. Keyfitz, B. L., and Kranzer, H. C. (1995). Spaces of weighted measures for conservation laws with singular solutions. J. Differential Equations 118, 420-451.

    Google Scholar 

  4. Keyfitz, B. L. (1999). Conservation laws, delta shocks and singular shocks. In Grosser, M., Hormann, G., Kunzinger, M., and Oberguggenberger, M. (eds.). Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC Press, Boca Raton, pp. 99-111.

    Google Scholar 

  5. KranzerH. C., and Keyfitz, B. L. (1990). A strictly hyperbolic system of conservation laws admitting singular shocks. In Keyfitz, B., and Shearer, M. (eds.), Nonlinear Evolution Equations that Change Type, IMA Math. Appl., Vol 27, Springer-Verlag, Berlin/New York, pp. 107-125.

    Google Scholar 

  6. Lax, P. D. (1971). Shock waves and entropy. In Zarontonello, E. H. (ed.), Proc. Symp. Univ. of Wisconsin, Academic Press, New York, pp. 603-634.

    Google Scholar 

  7. Liu, T. P., and Xin, Z. (1990). Overcompressive shock waves. In Keyfitz, B., and Shearer, M. (eds.), Nonlinear Evolution Equations that Change Type, IMA Math. Appl., Vol 27, Springer-Verlag, Berlin/New York, pp. 139-145.

    Google Scholar 

  8. Mock, M. (1980). A topological degree for orbits connecting critical points of autonomous systems. J. Differential Equations 38, 176-191.

    Google Scholar 

  9. Prue, H. P., and Sanders, R. (1987). On travelling-wave solutions to systems of conservation laws with singular viscosity. Com. P.D.E. 12, 1285-1307.

    Google Scholar 

  10. Sanders, R., and Sever, M. Computations with singular shocks. SIAM J. Appl. Dynamical Systems. (to appear).

  11. Sever, M. (2002). Viscous structure of singular shocks. Nonlinearity 15, 705-725.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, 77204-3008

    Richard Sanders

  2. Department of Mathematics, Hebrew University, Jerusalem, Israel

    Michael Sever

Authors
  1. Richard Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Sever
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sanders, R., Sever, M. The Numerical Study of Singular Shocks Regularized by Small Viscosity. Journal of Scientific Computing 19, 385–404 (2003). https://doi.org/10.1023/A:1025320412541

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025320412541

Search

Navigation