Type: Article
Publication Date: 1984-10-01
Citations: 16
DOI: https://doi.org/10.2307/2008281
We study the Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u, + fx(u) = 0, where /(h) is, say, strictly convex, /> ¿" > 0. We show that the divided differences of the numerical solution at time t do not exceed 2(/¿,)-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.Studying conservative difference approximations to (1.2), one aims at having (i) compactness, (ii) entropy condition.By compactness we merely mean the compactness of the family of solutions [v(-,t) = p(-,f; Ax),0< t < T,0< Ax^e).A standard tool used in that direction, e.g., [1], [3], [6], [11], is to guarantee that the total variation TV[t;(f)] = T,v\vv+l(t) -vv(t)\ remains bounded in time, v e LX(BV, [0, T]): since the mean value v(t) = £" v"(t)Ax is independent of t, it then