Abstract: We consider consistent, conservative-form, monotone finite difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the $BV$ (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin \cite{\WuYin} states that these so-called $BV$ entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase flow equation and hyperbolic conservation laws. The difference schemes are shown to converge to the unique $BV$ entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present context is to show that the approximations are $L^1$ Lipschitz continuous in the time variable (this is trivial for conservation laws). This continuity result is in turn intimately related to the regularity properties possessed by the (strongly degenerate) discrete diffusion term. We provide the necessary regularity estimates on the diffusion term by deriving and carefully analysing a linear difference equation satisfied by the numerical flux of the difference schemes.
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