We consider the initial value problem for degenerate
viscous and inviscid scalar conservation laws where the
flux function depends on the spatial location through a
"rough"coefficient function k(x).
We show that the Engquist-Osher
(and hence all monotone)
finite difference approximations converge
to the unique entropy solution
of the governing equation
if, among other demands, k' is in BV, thereby providing
alternative (new) existence proofs for entropy solutions of
degenerate convection-diffusion equations as
well as new convergence results for their finite difference
approximations. In the inviscid case, we also provide
a rate of convergence. Our convergence proofs are based on
deriving a series of a priori estimates
and using a general Lp compactness criterion.