  In this article we prove convergence of the Godunov scheme of [16] for a
scalar conservation law in one space dimension with a spatially discontinuous
flux. There may be infinitely many flux discontinuities, and the set of
discontinuities may have accumulation points. Thus the existence of traces
cannot be assumed. In contrast to the study appearing in [16], we do not
restrict the flux to be unimodal. We allow for the case where the flux has
degeneracies, i.e., the flux may vanish on some interval of state space. Since
the flux is allowed to be degenerate, the corresponding singular map may not be
invertible, and thus the convergence proof appearing in [16] does not pertain.
We prove that the Godunov approximations nevertheless do converge in the
presence of flux degeneracy, using an alternative method of proof. We
additionally consider the case where the flux has the form described in [21].
For this case we prove convergence via yet another method. This method of proof
provides a spatial variation bound on the solutions, which is of independent
interest. We present numerical examples that illustrate the theory.
