The Large-Time Behavior of the Scalar, Genuinely Nonlinear Lax-Friedrichs Scheme

We study the Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law ${u_t} + {f_x}(u) = 0$, where $f(u)$ is, say, strictly convex, $\ddot f \geqslant {\dot a_ \ast } > 0$. We show that the divided differences of the numerical solution at time t do not exceed $2{(t{\dot a_ \ast })^{ - 1}}$. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude, in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitative insight into the large-time behavior of the numerical computation.