Monotone difference approximations of BV solutions to degenerate convection-diffusion equations

We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we a llow the diffusion term to be strongly degenerate, solutions can be discont inuous and, in general, are 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 [Northeastern Math J., 5 (1989), pp. 395-422] states that these so-called BV entropy weak solutions are unique. The class of equations unde r consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation, and hyper bolic conservation laws. The difference schemes are shown to converge to th e unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the mai n difficulty in obtaining a similar convergence theory in the present conte xt is to show that the (strongly degenerate) discrete diffusion term is suf ficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satis ed by the numer ical flux of the difference schemes. Finally, we make some concluding remar ks about monotone difference schemes for multidimensional equations.