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Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure

• Benjamin Boutin1, , ,
• Frédéric Coquel2,  and
• Philippe G. LeFloch3,
• 1.Univ. Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
• 2.Centre de Mathématique Appliquées, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
• 3.Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Sorbonne Université, 75258 Paris, France
• * Corresponding author: Benjamin Boutin

Received:  October 2020

Revised:  January 2021

Early access:  February 2021

Published:  June 2021

The three authors were partially supported by the Innovative Training Networks (ITN) grant 642768 (ModCompShock), and by the Centre National de la Recherche Scientifique (CNRS)

Abstract

• In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.Keywords:Hyperbolic conservation law, coupling technique, Riemann problem, self-similar approximation, resonant effect. Mathematics Subject Classification: Primary: 35L65, 35D40.Citation: Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16(2): 283-315. doi: 10.3934/nhm.2021007📷

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Nonlinear Hyperbolic Systems - Mathematical Institute

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University of Oxford

https://www.maths.ox.ac.uk › oxpde-research-topics › n...

Apr 2, 2024 — The programme is toward mathematical research on nonlinear hyperbolic systems of conservation laws and related nonlinear problems, ...

Missing: dynamics ‎| Show results with: dynamics

The solution of non-linear hyperbolic equation systems by the finite element method

R. Löhner, K. Morgan, O. C. Zienkiewicz

First published: November 1984

https://doi.org/10.1002/fld.1650041105

Citations: 238

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Abstract The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co-ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor-Galerkin methods of Donea.17Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.

References

• Solution of the hyperbolic mild‐slope equation using the finite volume methodJ. Bokaris, K. AnastasiouInternational Journal for Numerical Methods in Fluids
• The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation lawsDawei Chen, Xijun Yu, Zhangxin ChenInternational Journal for Numerical Methods in Fluids
• High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equationMehdi Dehghan, Akbar MohebbiNumerical Methods for Partial Differential Equations
• A numerical method for solving the hyperbolic telegraph equationMehdi Dehghan, Ali ShokriNumerical Methods for Partial Differential Equations