Preprint 2001-009

A Priori and A Posteriori Error Estimates for the Staggered Lax-Friedrichs Scheme in Multi-Dimensions for Scalar Nonlinear Conservation Laws

Marc Küther

Abstract: Staggered grid finite volume methods (or also called {\em central schemes}) have been introduced in $1$d by Nessyahu and Tadmor in 1990 \cite{NT:90} in order to avoid the necessity to have information on solutions of Riemann problems for the evaluation of numerical fluxes. We consider the general case in multi dimensions and on general staggered grids which have to satisfy an overlap assumption, only. We interpret the staggered Lax-Friedrichs scheme as a three-step method consisting of a prolongation step onto a finer {\em intersection grid}, a finite volume step with an arbitrarily good numerical flux (e.g. Godunov flux) on the {\em intersection grid} followed by an averaging step such that the calculation of numerical fluxes reduce to evaluations of the continuous flux. Using this point of view, we prove an a posteriori error estimate and an a priori error estimate in the $L^1$ norm in space and time which is of order $h^{1/4}$ where $h$ is a mesh-size parameter. Hence, we recover for the staggered Lax-Friedrichs scheme the same order of convergence as for upwind finite volume methods on a fixed grid.

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Marc Küther, <>
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<> March 16 2001.

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