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, + 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. Introduction. We consider monotonicity-preserving schemes of the 3-point conservative form (1.1) vv(t + k) = vr(t) -\[h(vv(t), v,+1{t)) ä(*V-i(')> vv(t))], serving as consistent approximations to the scalar conservation law (1.2a) ^(x,t)+^(u(x,t)) = 0, and subject to initial data (1.2b) vv(t)\l=0 = u(x,,0), h(x,0) e L1 n r n £K. Here, vv(t) = v(xv, t) denotes the approximation value at the gridpoint (xv = vàx, t), k and àx are, respectively, the time-step and mesh size such that the mesh ratio X = k/Ax is kept fixed, and h(-, •) is the Lipschitz continuous numerical flux consistent with the differential one, h(v, v) = f(v). 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 Received May 26, 1983. 1980 Mathematics Subject Classification. Primary 65P05, 35L65. 'Research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-17070 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665. **Current Address: School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. ©1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 353 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use