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Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes

Published online by Cambridge University Press:  14 November 2011

Brian Hayes  and
Michael Shearer
Brian Hayes
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA
Michael Shearer
Affiliation:
Centre for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695–8205, USA
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Extract

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 733 - 754
Copyright
Copyright © Royal Society of Edinburgh 1999

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