This paper includes an erratum to "Convection and total variation flow", IMA J. Numer. Anal. 34, 1037-1071, 2014, that deals with a nonlinear hyperbolic scalar conservation law, regularised by the total variation flow operator (or 1-Laplacian), and in which a mistake has been written in the convergence proof of the numerical scheme to the continuous entropy solution. For correcting the proof, it is necessary to introduce an additional vanishing viscous term in the scheme. This modification imposes that the whole paper be cast in the framework of discrete and continuous solutions with unbounded support. This new version shows nevertheless a better result than the previous one, since the BV regularity and the compactness of the support of the initial data are no longer assumed.
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