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Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

Abstract

This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations.

We introduce a “kinetic relation” to act as a selection principle for choosing a unique non-classical solution to the Riemann problem. The convergence to non-classical weak solutions for the Cauchy problem is investigated. Using numerical experiments, we demonstrate that, for the cubic flux-function, the Beam-Warming scheme produces non-classical shocks while no such shocks are observed with the Lax-Wendroff scheme. All of these results depend crucially on the sign of the dispersion coefficient.

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(Accepted February 8, 1996)

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Hayes, B., LeFloch, P. Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws. Arch Rational Mech Anal 139, 1–56 (1997). https://doi.org/10.1007/s002050050046

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Keywords

  • Entropy
  • Weak Solution
  • Cauchy Problem
  • Dispersion Coefficient
  • Riemann Problem