Error Estimates for the Staggered Lax-Friedrichs Scheme on Unstructured Grids

Staggered grid finite volume methods (also called central schemes) were introduced in one dimension by Nessyahu and Tadmor in 1990 in order to avoid the necessity of having information on solutions of Riemann problems for the evaluation of numerical fluxes. We consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption. We interpret the staggered Lax–Friedrichs scheme as a three-step method consisting of a prolongation step onto a finer intersection grid , a finite volume step with an arbitrarily good numerical flux (e.g., Godunov flux) on the intersection grid , followed by an averaging step such that the calculation of numerical fluxes reduces 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 L1-norm in space and time which is of order h1/4, 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. AMS subject classifications. 65M15, 75M12, 35L65