In this paper, we prove the convergence of a finite-volume scheme for the time-dependent convection-diffusion equation with an L 1 right-hand side. To this purpose, we first prove estimates for the discrete solution and for its discrete time and space derivatives. Then we show the convergence of a sequence of discrete solutions obtained with more and more refined discretizations, possibly up to the extraction of a subsequence, to a function which mets the regularity requirements of the weak

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... ulation of the problem; to this purpose, we prove a compactness result, which may be seen as a discrete analogue to Aubin-Simon's lemma. Finally, such a limit is shown to be indeed a weak solution. 2000 Mathematics Subject Classification.