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 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.

中文翻译：

---

具有不连续通量的守恒定律的单调方案的收敛率

我们证明了一类具有不连续通量的标量守恒定律的单调有限体积方案在 $\mathrm{L}^1$ 中以 $\sqrt{\Delta x}$ 的速率收敛，只要通量在 u 中是严格单调的并且通量的空间依赖性是具有有限多个不连续性的分段常数。我们还提供了数值实验来说明主要结果。据我们所知，这是对具有不连续通量的守恒定律的数值方法的任何类型收敛速度的第一个证明。我们的证明依赖于具有初始值和边界值数据的守恒定律的收敛率。由于这些在文献中不容易获得，我们在附录中建立了这种情况下的收敛率。