Skip to main content
SpringerLink
Log in

Existence and stability of traveling discrete shocks

  • Conference paper
  • First Online:
Numerical Analysis and Its Applications (WNAA 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1196))

Included in the following conference series:

Abstract

We are concerned with existence and stability questions for finite difference schemes approximating solutions of scalar conservation laws with shocks. A suitable model for the study of the artifacts created by these schemes near the shocks are the traveling discrete shock profiles; these are discrete shock profiles υ=(υk)k∈ℤ which reappear shifted, when the scheme is applied on them, according to the speed of the shock. Existence of such profiles connecting entropy admissible shocks is already established for monotone schemes, first and third order accurate schemes and the Lax-Wendroff scheme. Jennings showed existence and l 1-stability of these profiles for conservative monotone schemes. Smyrlis showed existence and parametrization by the amount of excess mass and stability for stationary profiles of the Lax-Wendroff scheme. Shih Hsien Yu showed existence of traveling profiles of mild strength for the Lax-Wendroff scheme using inertial manifolds theory. Here we study traveling discrete shock profiles for Lax-Wendroff, Engquist-Osher and monotone schemes. We show existence of such profiles with small shock speed. We also show that these profiles are stable with respect to suitably weighted l 2-norms.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Courant, R., Friedrichs, K. O., Lewy, H., Über die Partiellen Differenzialgleichungen der Mathematischen Physik, Math. Ann., 100 (1928), pp. 32–74.

    Google Scholar 

  2. Björn Engquist, Shih Hsien Yu, Convergence of finite difference schemes for piecewise smooth solutions with shocks, to appear.

    Google Scholar 

  3. A. Harten, J.M. Hyman, P.D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math., 29 (1976), pp. 297–322.

    Google Scholar 

  4. A.M. Il'in, O.A. Oleinik, Behavior of the solution of the Riemann problem for certain quasilinear equations for unbounded increase of the time, Amer. Math. Soc. Transl. Ser.2, 42 (1964), pp. 19–23.

    Google Scholar 

  5. G. Jennings, Discrete shocks, Comm. Pure Appl. Math., 27 (1974), pp. 25–37.

    Google Scholar 

  6. C.K.R.T. Jones, R. Gardner, T. Kapitula, Stability of traveling waves for non-convex scalar viscous conservation laws, Comm. Pure Appl. Math., 46 (1993), pp. 505–526.

    Google Scholar 

  7. P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), pp. 537–566.

    Google Scholar 

  8. P.D. Lax, B. Wendroff, On Stability of finite difference schemes, Comm. Pure Appl. Math., 15 (1962), pp. 363–371.

    Google Scholar 

  9. Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Memoirs of the Amer. Math. Soc, Number 328, July 1985.

    Google Scholar 

  10. S. Kawashima, A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), pp. 97–127.

    Google Scholar 

  11. D. Michelson, Discrete shocks for difference approximations to system of conservation laws, Adv. Appl. Math., 5 (1984), pp. 433–469.

    Google Scholar 

  12. A. Majda, J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math., 32 (1979), pp. 445–482.

    Google Scholar 

  13. B. Engquist, S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), pp. 321–351.

    Google Scholar 

  14. D.H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), pp. 312–355.

    Google Scholar 

  15. Y.S. Smyrlis, Existence and Stability of Stationary Profiles of the LW Scheme, Comm. Pure. Appl. Math., 42 (1990), pp. 509–545.

    Google Scholar 

  16. Y.S. Smyrlis, Shih Hsien Yu, On the stability of traveling discrete shocks of conservation laws, to appear.

    Google Scholar 

  17. Shih Hsien Yu, Existence of Discrete Shock Profiles for the Lax-Wendroff Scheme, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus Nicosia, 1678, Cyprus

    Yiorgos Sokratis Smyrlis

  2. Department of Mathematics, University of California at Los Angeles, 90024, Los Angeles, CA

    Shih Hsien Yu

Authors
  1. Yiorgos Sokratis Smyrlis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shih Hsien Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Lubin Vulkov Jerzy Waśniewski Plamen Yalamov

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Smyrlis, Y.S., Yu, S.H. (1997). Existence and stability of traveling discrete shocks. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol 1196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62598-4_127

Download citation

  • DOI: https://doi.org/10.1007/3-540-62598-4_127

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62598-8

  • Online ISBN: 978-3-540-68326-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation