This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: (1) Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform $\operatorname{BV}$ bound on the numerical approximations; (2) Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; (3) Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.

中文翻译：

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非局部守恒定律耦合系统的适定性和误差估计


本文讨论了非局部双曲守恒定律耦合系统的熵解的数值近似的误差估计。这些系统可以通过对流项中存在的非局部系数进行强耦合。正在考虑一类相当通用的磁通量，其中磁通量的局部部分可以在无限多个点处不连续，并可能具有累积点。本文的目标有三个：（1） 通过在数值近似上推导出一个均匀的 $\operatorname{BV}$ 边界，确定此类系统存在具有粗糙局部通量的熵解;（2） 为具有平滑和粗糙局部磁通量的此类系统推导出一般的库兹涅佐夫型引理（因此是唯一性的）;（3） 证明有限体积近似与系统熵解的收敛速率为 $1/2$ 和 $1/3$，分别具有齐次（在任何维度中）和粗糙的局部部分（在一维中）。包括数值实验以说明收敛速率。