Nonlinear hyperbolic systems

1991 ◽  
pp. 110-130
Author(s):  
Michael E. Taylor
Keyword(s):  
Hyperbolic Systems ◽  
Nonlinear Hyperbolic Systems

On the Convergence Rate of Vanishing Viscosity Approximations for Nonlinear Hyperbolic Systems

2012 ◽  
Vol 44 (5) ◽  
pp. 3537-3563 ◽  
Author(s):  
Alberto Bressan ◽  
Feimin Huang ◽  
Yong Wang ◽  
Tong Yang
Keyword(s):  
Convergence Rate ◽  
Hyperbolic Systems ◽  
Vanishing Viscosity ◽  
Nonlinear Hyperbolic Systems

scholarly journals Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

2010 ◽  
Vol 229 (18) ◽  
pp. 6485-6511 ◽  
Author(s):  
J. Tryoen ◽  
O. Le Maître ◽  
M. Ndjinga ◽  
A. Ern
Keyword(s):  
Hyperbolic Systems ◽  
Galerkin Methods ◽  
Nonlinear Hyperbolic Systems

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.

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