Hyperbolic Systems with Discontinuous Coefficients: Generalized Wavefront Sets

2008 ◽  
pp. 117-136 ◽  
Author(s):  
Michael Oberguggenberger
Keyword(s):  
Hyperbolic Systems ◽  
Discontinuous Coefficients

scholarly journals HAAR METHOD, AVERAGED MATRIX, WAVE CANCELLATIONS, AND L1STABILITY FOR HYPERBOLIC SYSTEMS

2006 ◽  
Vol 03 (04) ◽  
pp. 701-739
Author(s):  
PHILIPPE G. LEFLOCH
Keyword(s):  
Hyperbolic Systems ◽  
Front Tracking ◽  
Discontinuous Coefficients ◽  
Entropy Solutions ◽  
Main Difficulty ◽  
Discontinuous Solutions ◽  
Type Method ◽  
Main Observation ◽  
Fluid Dynamics Equations ◽  
Nonlinear Hyperbolic Systems

We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks — which are recognized as a source for non-uniqueness and instability. Our Haar–Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It generalizes the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1norm upon their initial data, by exhibiting an L1functional controlling the distance between two solutions.

15.—The Propagation of Second-order Lipschitz Discontinuities in Quasi-linear Hyperbolic Systems with Discontinuous Coefficients

1976 ◽  
Vol 74 ◽  
pp. 205-224
Author(s):  
Alan Jeffrey ◽  
Esin Inan
Keyword(s):  
General Theory ◽  
Hyperbolic Systems ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
Strong Discontinuity ◽  
Plane Shear ◽  
Reflected Waves ◽  
Weak Discontinuities ◽  
Derivatives Of ◽  
Transmission And Reflection

SynopsisThis paper develops the general theory of the propagation of Lipschitz discontinuities in first- and second-order partial derivatives of the initial data for a conservative quasi-linear hyperbolic system with discontinuous coefficients. After establishing that such weak discontinuities propagate along characteristics the appropriate transport equations are derived. The effect on this form of wave propagation of the strong discontinuity associatedwith the discontinuous coefficients is then studied and the transmission and reflection characteristics of the resulting waves are analysed. In conclusion, an application of this general theory is made to the propagation of plane shear waves through two different continuous hyperelastic solids to determine the transmitted and reflected waves.

The propagation of weak discontinuities in quasi-linear hyperbolic systems when a characteristic shock occurs

1978 ◽  
Vol 78 (3-4) ◽  
pp. 285-290 ◽  
Author(s):  
Andrea Donato
Keyword(s):  
Hyperbolic Systems ◽  
Discontinuous Coefficients ◽  
Strong Discontinuity ◽  
Systems Of Equations ◽  
Hyperbolic Systems Of Equations ◽  
Characteristic Shock ◽  
Weak Discontinuities ◽  
Speed Of Propagation

SynopsisIn this paper we study the propagation of weak discontinuities in quasi-linear hyperbolic systems of equations with discontinuous coefficients when one or more speeds of propagation o f the discontinuity wave is coincident with the speed of propagation of the strong discontinuity.

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