An error estimate for the finite difference approximation to degenerate   convection -diffusion equations
  We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection-diffusion equations in one space dimension, and prove an L1 error estimate. Precisely, we show that the L1 loc difference between the approximate solution and the unique entropy solution converges at a rate O(\Deltax 1/11), where \Deltax is the spatial mesh size. If the diffusion is linear, we get the convergence rate O(\Deltax 1/2), the point being that the O is independent of the size of the diffusion 
K. H. Karlsen, U. Koley, N. H. Risebro
math.AP