Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations

TLDR: This work considers consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension and provides the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes.

Abstract: We consider consistent, conservative-form, monotone 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, 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 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 (strongly degenerate) discrete diffusion term is sufficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes. Finally, we make some concluding remarks about monotone difference schemes for multidimensional equations.