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)  in the context of constant coefficient linear advection equations. © 2011 Elsevier B.V. All rights reserved.
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