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We study the Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law      u t + f x ( u ) = 0 {u_t} + {f_x}(u) = 0      , where      f ( u ) f(u)       is, say, strictly convex,      f ¨  ⩾  a ˙  ∗  &gt; 0 \ddot f \geqslant {\dot a_ \ast } &gt; 0      . We show that the divided differences of the numerical solution at time  t  do not exceed      2 ( t a ˙  ∗  ) −  1 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.

The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme