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The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws

1997; Society for Industrial and Applied Mathematics; Volume: 34; Issue: 6 Linguagem: Inglês

10.1137/s003614299529347x

1095-7170

Florin Şabac,

Fluid Dynamics and Turbulent Flows

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.