The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws

We are interested in the rate of convergence in L-1 of the approximate solution of a conservation law generated by a monotone finite difference scheme, Kuznetsov has proved that this rate is 1/2 [USSR Comput. Math, Math. Phys., 16 (1976), pp. 105-119 and Topics Numer. Anal. III, in Proc. Roy. Irish Acad. Conf., Dublin, 1976, pp. 133-197], and recently Teng: and Zhang have proved this estimate to be sharp for a linear flux [SIAM J. Numer. Anal., 34 (1997), pp. 959-978]. We prove, by constructing appropriate initial data for the Cauchy problem, that Kuznetsov's estimates are sharp for a nonlinear flux as well.