Finite volume schemes for locally constrained conservation laws |
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Authors: | Boris Andreianov Paola Goatin Nicolas Seguin |
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Institution: | 1. Laboratoire de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030, Besan?on Cedex, France 2. ISITV, Université du Sud Toulon-Var, Avenue Georges Pompidou, BP 56, 83162, La Valette du Var Cedex, France 3. Laboratoire J.-L. Lions, UPMC Univ Paris 06, BC 187, 75252, Paris Cedex 05, France
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Abstract: | This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin (J. Differ. Equ.
234(2):654–675, 2007). The model involves a standard conservation law supplemented by a local unilateral constraint on the
flux at the point x = 0 (modelling a road light, a toll gate, etc.). We first show that the problem can be interpreted in terms of the theory
of conservation laws with discontinuous flux function, as developed by Adimurthi et al. (J. Hyperbolic Differ. Equ. 2(4):783–837,
2005) and Bürger et al. (SIAM J. Numer. Anal. 47(3):1684–1712, 2009). We reformulate accordingly the notion of entropy solution
introduced by Colombo and Goatin (J. Differ. Equ. 234(2):654–675, 2007), and extend the well-posedness results to the L
∞ framework. Then, starting from a general monotone finite volume scheme for the non-constrained conservation law, we produce
a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process solution.
Numerical examples modelling a “green wave” are presented. |
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