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Numerical methods with controlled dissipation for small-scale dependent shocks

By Philippe G. LeFloch and Siddhartha Mishra

Abstract

72 pagesWe provide a 'user guide' to the literature of the past twenty years concerning the modeling 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, 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 the general well-posedness theory, small-scale de- pendent 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

Topics: [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP]
Publisher: HAL CCSD
Year: 2013
OAI identifier: oai:HAL:hal-00916932v1
Provided by: Hal-Diderot

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