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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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References

1Allen, M. B. III, Behie, G. A. and Trangenstein, J. A.. Multiphase flow in porous media: mechanics, mathematics and numerics (Springer, 1988).Google Scholar
2Azevedo, A. V., Marchesin, D., Plohr, B. and Zumbmn, K.. Bifurcation of nonclassical viscous shock profiles from the constant state. Commun. Math. Phys. (In the press.)Google Scholar
3Guckenheimer, J. and Holmes, P.. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer, 1983).Google Scholar
4Hayes, B. T. and LeFloch, P. G.. Nonclassical shock waves: scalar conservation laws. Arch. Ration. Mech. Analysis 139 (1997), 1–56.Google Scholar
5Hayes, B. T. and LeFLoch, P. G.. Nonclassical shock waves and kinetic relations: finite difference schemes. SIAM J. Numer. Analysis 35 (1998), 2169–2194.Google Scholar
6Isaacson, E., Marchesin, D. and Plohr, B.. Transitional waves for conservation laws. SIAM J. Math. Analysis 21 (1990), 837–866.Google Scholar
7Jacobs, D., McKinney, W. and Shearer, M.. Travelling wave solutions of the modified Korteweg–de Vries–Burgers equation. J. Diff. Eqns 116 (1995), 448–467.Google Scholar
8Lax, P. D.. Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10 (1957), 537–566.Google Scholar
9Lax, P. D. and Wendroff, B.. Systems of conservation laws. Commun. Pure Appl. Math. 13 (1960), 217–237.Google Scholar
10LeVeque, R. J.. Numerical methods for conservation laws (Birkhöuser, 1990).Google Scholar
11Oleinik, O.. Uniqueness and stability of the generalized solution of the Cauchy problemfor a quasilinear equation. Usp. Mat. Nauk. 14 (1959), 165–170. (English transl. Am. Math. Soc. Transl. 2 33 (1964), 285–290.)Google Scholar
12Schecter, S.. Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral. Nonlinearity 3 (1990), 79–99.Google Scholar
13Shearer, M. and Yang, Y.. The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity. Proc. R. Soc. Edinb. A 125 (1995), 675–699.Google Scholar
14Slemrod, M.. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Analysis 81 (1983), 301–315.Google Scholar
15Wu, C. C.. New theory of MHD shock waves. Viscous profiles and numerical methods for shock waves (ed. Shearer, M.) (SIAM, 1991).Google Scholar