An Introduction to Nonclassical Shocks of Systems of Conservation Laws

  • Philippe G. LeFloch
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 5)


We review a recent activity on nonclassical shock waves of strictly hyperbolic systems of conservation laws, generated by balanced diffusion and dispersion effects. These shocks do not satisfy the standard Lax and Liu entropy criteria, and in fact are undercompressive and satisfy a single entropy inequality. The selection of admissible nonclassical shocks requires a strengthened version of the entropy inequality, called a kinetic relation, which constrains the entropy dissipation. The kinetic function is determined from traveling wave solutions to a system of equations augmented with diffusion and dispersion.

For nonconvex scalar conservation laws and non-genuinely nonlinear, strictly hyperbolic systems, the existence and uniqueness of nonclassical shocks is investigated using successively the traveling wave analysis, the front tracking algorithm and the compensated compactness method. Nonclassical shocks may also be generated by finite difference schemes.

The kinetic relation provides a useful tool to study the properties of nonclassical shocks and, in particular, their sensitivity to regularization parameters.


Travel Wave Solution Riemann Problem Entropy Inequality Shock Speed Riemann Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Philippe G. LeFloch
    • 1
  1. 1.Centre de Mathématiques Appliquées & Centre National de la Recherche ScientifiqueUA 756, Ecole PolytechniquePalaiseau CedexFrance

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