Error estimate for the Finite Volume Scheme applied to the advection equation

We study the convergence of a Finite Volume scheme for the linear advection equation with a Lipschitz divergence-free speed in $\R^d$. We prove a $h^{1/2}$-error estimate in the $L^\infty(0,t;L^1)$-norm for $BV$ data. This result was expected from numerical experiments and is optimal.