scholarly journals Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

1999 ◽  
Vol 68 (227) ◽  
pp. 955-971 ◽  
Author(s):  
A. Chalabi
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
Relaxation Schemes

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.

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