Error bounds for the large time step Glimm scheme applied to scalar conservation laws

In this paper we derive an \(L^1\) error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by \(O(\Delta x^{1/2}\vert \log\Delta x\vert )\) for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2.

Authors and Affiliations

1. Beijing University of Chemical Technology, Beijing 100029, People's Republic of China; e-mail: huangjy@buct.edu.cn , , , , , , CN

   Jinyang Huang

2. Institute of Systems Science, Academia Sinica, Beijing 100080, People's Republic of China; e-mail: jwang@iss06.iss.ac.cn , , , , , , CN

   Jinghua Wang

3. Institut für Analysis und Numerik, Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany; e-mail: gerald.warnecke@mathematik.uni-magdeburg.de , , , , , , DE

   Gerald Warnecke

Received April 10, 2000 / Revised version received January 16, 2001 / Published online September 19, 2001

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