We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and the existence of one strictly convex entropy. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case ($k\equiv 1$).

翻译：我们建立了用于求解非线性守恒律的有限差分数组的定量紧致性估计。这些方程涉及通量函数 $f(k(x,t),u)$，其中系数 $k(x,t)$ 是 $BV$ 正则的，并可能在 $(x,t)$ 平面沿曲线出现间断。我们的方法在技术上具有初等性，依赖于离散相互作用估计和一个严格凸熵的存在性。虽然具体细节以 Lax-Friedrichs 格式为例进行阐述，但同一框架可应用于其他差分数组。值得注意的是，即使在齐次情形 ($k\equiv 1$) 下，我们的紧致性估计也是全新的。