# Preprint 2012-022

# Error estimate for Godunov approximation of locally constrained conservation laws

## Clément Cancès and Nicolas Seguin

**Abstract:**
We consider a model of traffic flow
with unilateral constraint on the flux
introduced by R. M. Colombo and P. Goatin [14],
for which the convergence of numerical approximation
using monotone finite volume schemes
has been performed by B. Andreianov et al. [4].
We derive for this problem some new 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.

**References**

[4] B. Andreianov, P. Goatin, and N. Seguin,

*Finite volume schemes for locally constrained conservation laws*, Numer. Math.,

**115**(2010), pp. 609–645. With supplementary material available online.

[14] R. M. Colombo and P. Goatin,

*A well posed conservation law with a variable unilateral constraint*, J. Differential Equations,

**234**(2007), pp. 654–675.