Discrete entropy inequality

For a scalar conservative monotone update with flux $F$, define $Q(a,b;c)=F(a\vee c,b\vee c)-F(a\wedge c,b\wedge c)$. Monotonicity and preservation of constant states give $|U_j^{n+1}-c|\le |U_j^n-c|-(k/d)(Q_{j+1/2}-Q_{j-1/2})$. At equal states $Q(u,u;c)=\mathrm{sgn}(u-c)(f(u)-f(c))$, the Kruzhkov entropy flux. Such discrete inequalities connect stability of a monotone conservative scheme with the admissibility of its limiting entropy solution.