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Numerical computation of viscous profiles for hyperbolic conservation laws

Authors: Heinrich Freistühler and Christian Rohde
Journal: Math. Comp. 71 (2002), 1021-1042
MSC (2000): Primary 65L10; Secondary 35L65, 34C37, 76W05
Published electronically: October 26, 2001
MathSciNet review: 1898744
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Abstract: Viscous profiles of shock waves in systems of conservation laws can be viewed as heteroclinic orbits in associated systems of ordinary differential equations (ODE). In the case of overcompressive shock waves, these orbits occur in multi-parameter families. We propose a numerical method to compute families of heteroclinic orbits in general systems of ODE. The key point is a special parameterization of the heteroclinic manifold which can be understood as a generalized phase condition; in the case of shock profiles, this phase condition has a natural interpretation regarding their stability. We prove that our method converges and present numerical results for several systems of conservation laws. These examples include traveling waves for the Navier-Stokes equations for compressible viscous, heat-conductive fluids and for the magnetohydrodynamics equations for viscous, heat-conductive, electrically resistive fluids that correspond to shock wave solutions of the associated ideal models, i.e., the Euler, resp. Lundquist, equations.

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Additional Information

Heinrich Freistühler
Affiliation: Max–Planck–Institut für Mathematik in den Naturwissenschaften, Inselstr. 22-26, D-04103 Leipzig, Germany

Christian Rohde
Affiliation: Institut für Angewandte Mathematik, Albert–Ludwigs–Universtät Freiburg, Hermann–Herder–Str. 10, D-79104 Freiburg, Germany

Keywords: Shock waves, connecting heteroclinic manifolds, boundary value problems for ODE, magnetohydrodynamics
Received by editor(s): December 16, 1998
Received by editor(s) in revised form: September 12, 2000
Published electronically: October 26, 2001
Additional Notes: The authors acknowledge support by the DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme” and by the EU-TMR research network for Hyperbolic Conservation Laws (project # ERBFMRXCT960033).
Article copyright: © Copyright 2001 American Mathematical Society