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Numerical methods with controlled dissipation for small-scale dependent shocks*

Published online by Cambridge University Press:  12 May 2014

Philippe G. LeFloch  and
Siddhartha Mishra
Philippe G. LeFloch
Affiliation:
Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75258 Paris, France, E-mail: contact@philippelefloch.org
Siddhartha Mishra
Affiliation:
Seminar for Applied Mathematics, D-Math, Eidgenössische Technische Hochschule, HG Raemistrasse, Zürich–8092, Switzerland, E-mail: siddhartha.mishra@sam.math.ethz.ch
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Extract

We provide a ‘user guide’ to the literature of the past twenty years concerning the modelling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, and boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with general well-posedness theory, small-scale dependent solutions are characterized by a kinetic relation, a family of paths, or an admissible boundary set. We provide a review of numerical methods (front-tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically meaningful dissipation mechanisms of interest in the applications. An essential role is played by the equivalent equation associated with discrete schemes, which is found to be relevant even for solutions containing shock waves.

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Type
Research Article
Information
Acta Numerica , Volume 23 , May 2014 , pp. 743 - 816
Copyright
Copyright © Cambridge University Press 2014 

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