In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel $\mu$ or the flux $f$. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of $\sqrt{\Delta t}$ in $L^1(\mathbb{R})$. To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.

翻译：本文讨论了一类单调有限体积格式在近似非局部标量守恒律时的误差分析，这类守恒律用于模拟交通流和人群动力学，且未对核函数$\mu$或通量$f$的单调性或线性性施加额外假设。我们首先证明了此类偏微分方程的一个新型库兹涅佐夫型引理，并进而表明有限体积近似在$L^1(\mathbb{R})$范数下以$\sqrt{\Delta t}$的速率收敛到熵解。据我们所知，这是针对此类守恒律的首个收敛速率证明。我们还通过数值实验验证了这一结果。