Structure-Preserving Shock-Capturing Methods: Late-Time Asymptotics, Curved Geometry, Small-Scale Dissipation, and Nonconservative Products

Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 3)

Abstract

We consider weak solutions to nonlinear hyperbolic systems of conservation laws arising in compressible fluid dynamics and we describe recent work on the design of structure-preserving numerical methods. We focus on preserving, on one hand, the late-time asymptotics of solutions and, on the other hand, the geometrical effects that arise in certain applications involving curved space. First, we study here nonlinear hyperbolic systems with stiff relaxation in the late time regime. By performing a singular analysis based on a Chapman–Enskog expansion, we derive an effective system of parabolic type and we introduce a broad class of finite volume schemes which are consistent and accurate even for asymptotically late times. Second, for nonlinear hyperbolic conservation laws posed on a curved manifold, we formulate geometrically consistent finite volume schemes and, by generalizing the Cockburn–Coquel–LeFloch theorem, we establish the strong convergence of the approximate solutions toward entropy solutions.

Notes

Acknowledgements

The author was partially supported by the Agence Nationale de la Recherche (ANR) through the grant ANR SIMI-1-003-01, and by the Centre National de la Recherche Scientifique (CNRS). These notes were written at the occasion of a short course given by the authors at the University of Malaga for the XIV Spanish-French School Jacques-Louis Lions. The author is particularly grateful to C. Vázquez-Cendón and C. Parés for their invitation, warm welcome, and efficient organization during his stay in Malaga.

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsCentre National de la Recherche Scientifique and Université Pierre et Marie CurieParisFrance

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