Abstract: We establish $L^1$ convergence of a viscous splitting method for nonlinear possibly strongly degenerate convection-diffusion problems. Since we allow the equations to be strongly degenerate, solutions can be discontinuous and they are not, in general, uniquely determined by their data. We thus consider entropy weak solutions realized by the vanishing viscosity method. This notion is broad enough to also include non-degenerate parabolic equations as well as hyperbolic conservation laws. It thus provides a suitable ``$L^1$ type'' framework for analyzing numerical schemes for convection-diffusion problems that are designed to handle various balances of convective and diffusive forces. We present a numerical example which shows that our splitting scheme has such ``design''.
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