Degenerate convection-diffusion equations and implicit monotone difference schemes

We analyse implicit monotone finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion equations in one spatial dimension. Since we allow strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. We thus choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. This paper complements our previous work [8] on explicit monotone schemes.