Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

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Abstract

In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations. © 2011 Elsevier B.V. All rights reserved.

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Bouche, D., Ghidaglia, J. M., & Pascal, F. P. (2011). Error estimate for the upwind finite volume method for the nonlinear scalar conservation law. Journal of Computational and Applied Mathematics, 235(18), 5394–5410. https://doi.org/10.1016/j.cam.2011.05.050

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