Convection and total variation flow-erratum and improvement

摘要

This paper includes an erratum to (Bouchut et al. (2014) Convection and total variation flow. IMA J. Numer. Anal., 34, 1037-1071.) which deals with a nonlinear hyperbolic scalar conservation law, regularized by the total variation flow operator (or 1-Laplacian), and in which a mistake occurred 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 requires casting the whole paper in the framework of discrete and continuous solutions with unbounded support. This new version, nevertheless, leads to a better result than the previous one, since the bounded variation regularity and the compactness of the support of the initial data are no longer assumed.