Finite volume schemes for locally constrained conservation laws

This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin in [CG07]. 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. [AMG05] and Bürger et al. [BKT09]. We reformulate accordingly the notion of entropy solution introduced in [CG07], 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.