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 entropy function. 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：我们针对用于求解非线性守恒律的有限差分格式建立了定量紧性估计。这些方程涉及通量函数$f(k(x,t),u)$，其中系数$k(x,t)$具有$BV$正则性，并可能在$(x,t)$平面上沿曲线出现间断。我们的方法在技术层面上较为初等，它依赖于离散交互估计与一个熵函数的存在性。虽然具体细节以Lax-Friedrichs格式为例进行阐述，但相同框架可应用于其他差分格式。值得注意的是，即使在齐次情形（$k\equiv 1$）下，我们的紧性估计也是全新的。