  We prove that a class of monotone finite volume schemes for scalar
conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta
x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the
spatial dependency of the flux is piecewise constant with finitely many
discontinuities. We also present numerical experiments to illustrate the main
result. To the best of our knowledge, this is the first proof of any type of
convergence rate for numerical methods for conservation laws with
discontinuous, nonlinear flux. Our proof relies on convergence rates for
conservation laws with initial and boundary value data. Since those are not
readily available in the literature we establish convergence rates in that case
en passant in the Appendix.
