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

Summary.

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.