An optimal error estimate for upwind Finite Volume methods for nonlinear hyperbolic conservation laws

Authors:
Daniel Bouche;Jean-Michel Ghidaglia;FréDéRic P. Pascal
Affiliations:
Commissariat í línergie Atomique (CEA DAM), DIF, F-91297 Arpajon, France and CMLA, ENS Cachan et CNRS, UniverSud, 61 avenue du Président Wilson, F-94235 Cachan, France and LRC MESO, ...;CMLA, ENS Cachan et CNRS, UniverSud, 61 avenue du Président Wilson, F-94235 Cachan, France and LRC MESO, ENS de Cachan, CEA DAM, 61 avenue du Président Wilson, F-94235 Cachan, France;CMLA, ENS Cachan et CNRS, UniverSud, 61 avenue du Président Wilson, F-94235 Cachan, France and LRC MESO, ENS de Cachan, CEA DAM, 61 avenue du Président Wilson, F-94235 Cachan, France
Venue:
Applied Numerical Mathematics
Year:
2011

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Abstract

The purpose of this paper is to show that the cell-centered upwind Finite Volume scheme applied to general hyperbolic systems of m conservation laws approximates smooth solutions to the continuous problem at order one in space and time. As it is now well understood, there is a lack of consistency for order one upwind Finite Volume schemes: the truncation error does not tend to zero as the time step and the grid size tend to zero. Here, following our previous papers on scalar equations, we construct a corrector that allows us to prove the expected error estimate for nonlinear systems of equations in one dimension.