  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 ($k\equiv 1$).
