Numerical Analysis
Transport-equilibrium schemes for computing nonclassical shocks
Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 623-626.

This Note presents a very efficient numerical strategy for computing weak solutions of a scalar conservation law which fails to be genuinely nonlinear. In such a situation, the dynamics of shock solutions turns out to be mainly driven by a prescribed kinetic function that imposes the speed of propagation of the discontinuities. We show how to enforce the validity of the kinetic criterion at the discrete level. The resulting scheme provides, in addition, sharp profiles. Numerical evidence are included.

Cette Note présente un algorithme très efficace pour le calcul des solutions faibles d'une loi de conservation scalaire non vraiment nonlinéaire. Dans ce contexte, la dynamique des solutions choc repose principalement sur la donnée d'une fonction cinétique qui fixe la vitesse de propagation des discontinuités. Nous montrons comment forcer la validité du critère cinétique au niveau discret. Le schéma obtenu fournit par ailleurs des discontinuités sans diffusion numérique. Des résultats numériques sont présentés.

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DOI: 10.1016/j.crma.2006.02.003
Chalons, Christophe 1

1 Laboratoire J.-L Lions, U.M.R. 7598, université P. et M. Curie, boîte courrier 187, 75252 Paris cedex 05, France
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Chalons, Christophe. Transport-equilibrium schemes for computing nonclassical shocks. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 623-626. doi : 10.1016/j.crma.2006.02.003. http://www.numdam.org/articles/10.1016/j.crma.2006.02.003/

[1] C. Chalons, Transport-Equilibrium schemes for computing nonclassical shocks. I. Scalar conservation laws, Laboratoire J.-L. Lions, Preprint R05031, 2005. Available at http://www.ann.jussieu.fr

[2] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows, Laboratoire J.-L. Lions, Preprint R05032, 2005. Available at http://www.ann.jussieu.fr

[3] C. Chalons, Transport-equilibrium schemes for computing nonclassical shocks, II. A hyperbolic-elliptic model of phase transitions, in preparation

[4] LeFloch, P.G. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, E.T.H. Lecture Notes Series, Birkhäuser, 2002

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