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Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  08 March 2017

Abdelaziz Beljadid ,
Philippe G. LeFloch ,
Siddhartha Mishra  and
Carlos Parés
Abdelaziz Beljadid*
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Philippe G. LeFloch*
Affiliation:
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France
Siddhartha Mishra*
Affiliation:
Seminar for Applied Mathematics (SAM), ETH Zurich, Rämistrasse-101, Zürich, 8092, Switzerland
Carlos Parés*
Affiliation:
Departamento de Análisis Matemático, Universidad de Málaga, 29071 Málaga, Spain
*
*Corresponding author. Email addresses:beljadid@mit.edu (A. Beljadid), contact@philippelefloch.org (P. G. LeFloch), siddhartha.mishra@sam.math.ethz.ch (S. Mishra), pares@uma.es (C. Parés)
*Corresponding author. Email addresses:beljadid@mit.edu (A. Beljadid), contact@philippelefloch.org (P. G. LeFloch), siddhartha.mishra@sam.math.ethz.ch (S. Mishra), pares@uma.es (C. Parés)
*Corresponding author. Email addresses:beljadid@mit.edu (A. Beljadid), contact@philippelefloch.org (P. G. LeFloch), siddhartha.mishra@sam.math.ethz.ch (S. Mishra), pares@uma.es (C. Parés)
*Corresponding author. Email addresses:beljadid@mit.edu (A. Beljadid), contact@philippelefloch.org (P. G. LeFloch), siddhartha.mishra@sam.math.ethz.ch (S. Mishra), pares@uma.es (C. Parés)
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Abstract

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic systems in nonconservative form—the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

Keywords

Nonlinear hyperbolicnonconservative systemdiscrete schemecontrolled dissipationcomplex fluid dynamics

MSC classification

Secondary: 35L65: Conservation laws 65M06: Finite difference methods 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 913 - 946
Copyright
Copyright © Global-Science Press 2017 

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