scholarly journals On the structure of BV entropy solutions for hyperbolic systems of balance laws with general flux function

2019 ◽  
Vol 16 (02) ◽  
pp. 333-378
Author(s):  
Fabio Ancona ◽  
Laura Caravenna ◽  
Andrea Marson
Keyword(s):  
Conservation Laws ◽  
Wave Front ◽  
Hyperbolic System ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Systems ◽  
Entropy Solutions ◽  
Balance Laws ◽  
Hyperbolic Conservation ◽  
Flux Function ◽  
Systems Of Balance Laws

The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results in [S. Bianchini and L. Yu, Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension, Comm. Partial Differential Equations 39(2) (2014) 244–273] for strictly hyperbolic system of conservation laws.

scholarly journals Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

2015 ◽  
Vol 12 (01) ◽  
pp. 189-219
Author(s):  
Alexey Miroshnikov ◽  
Konstantina Trivisa
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Stability And Convergence ◽  
Balance Laws ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
Hyperbolic Balance Laws ◽  
Relaxation Schemes ◽  
Systems Of Balance Laws ◽  
Weak Dissipation

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

scholarly journals On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Ciro D'Apice ◽  
Peter I. Kogut ◽  
Rosanna Manzo
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Dynamic Models ◽  
Optimization Problems ◽  
Fluid Dynamic ◽  
Entropy Solutions ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
Linear Evolution ◽  
Linear Evolution Equation

We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.

scholarly journals Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

2018 ◽  
Vol 13 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Eduardo Abreu ◽  
◽  
John Perez ◽  
Arthur Santo ◽  
◽  
...  
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Approximation Methods ◽  
Balance Laws ◽  
Hyperbolic Conservation

scholarly journals Optimal jump set in hyperbolic conservation laws

2020 ◽  
Vol 17 (04) ◽  
pp. 765-784
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Hyperbolic Systems ◽  
Scalar Case ◽  
Qualitative Properties ◽  
Hyperbolic Conservation ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Jump Set

We investigate qualitative properties of entropy solutions to hyperbolic conservation laws, and construct an entropy solution to a scalar conservation law for which the jump set is not closed, in particular, it is dense in a space-time domain. In a second part, we establish a similar result for hyperbolic systems. We give two different approaches for scalar conservation laws and hyperbolic systems in order to obtain these results. For the scalar case, the solutions are explicitly calculated.

Basic structures of hyperbolic relaxation systems

2002 ◽  
Vol 132 (5) ◽  
pp. 1259-1274 ◽  
Author(s):  
Wen-An Yong
Keyword(s):  
Small Parameter ◽  
Conservation Laws ◽  
Structural Properties ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Systems ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
First Order ◽  
Relaxation Systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

Sign in / Sign up

Export Citation Format

Share Document