The Optimal Convergence Rate for Monotone Finite Difference Schemes
SIAM Journal on Numerical Analysis Vol. 34, No. 6 (Dec., 1997), pp. 2306-2318
University of Alberta School of Business Research Paper No. 2013-1097
Posted: 2 Jul 2013
Date Written: June 1, 1996
Abstract
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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