THE OPTIMAL CONVERGENCE RATE OF MONOTONE FINITE DIFFERENCE METHODS FOR HYPERBOLIC CONSERVATION LAWS∗

We are interested in the rate of convergence in L1 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. 183–197], 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.

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