Error Estimate for Godunov Approximation of Locally Constrained Conservation Laws

Abstract

We consider a model of traffic flow with unilateral constraint on the flux introduced by Colombo and Goatin [J. Differential Equations, 234 (2007), pp. 654--675], for which the convergence of numerical approximation using monotone finite volume schemes has been performed by Andreianov, Goatin, and Seguin [Numer. Math., 115 (2010), pp. 609--645]. We derive for this problem a new ${\rm BV}$ estimate and make use of it to provide an error estimate for the Godunov approximation of the problem of order $h^{1/3}$ that is improved into the optimal order $h^{1/2}$ under a reasonable assumption. Numerical experiments are then provided to illustrate the optimality of the result.