  In this paper, we focus on the finite difference approximation of nonlinear
degenerate parabolic equations, a special class of parabolic equations where
the viscous term vanishes in certain regions. This vanishing gives rise to
additional challenges in capturing sharp fronts, beyond the restrictive CFL
conditions commonly encountered with explicit time discretization in parabolic
equations. To resolve the sharp front, we adopt the high-order multi-resolution
alternative finite difference WENO (A-WENO) methods for the spatial
discretization. To alleviate the time step restriction from the nonlinear stiff
diffusion terms, we employ the exponential time differencing Runge-Kutta
(ETD-RK) methods, a class of efficient and accurate exponential integrators,
for the time discretization. However, for highly nonlinear spatial
discretizations such as high-order WENO schemes, it is a challenging problem
how to efficiently form the linear stiff part in applying the exponential
integrators, since direct computation of a Jacobian matrix for high-order WENO
discretizations of the nonlinear diffusion terms is very complicated and
expensive. Here we propose a novel and effective approach of replacing the
exact Jacobian of high-order multi-resolution A-WENO scheme with that of the
corresponding high-order linear scheme in the ETD-RK time marching, based on
the fact that in smooth regions the nonlinear weights closely approximate the
optimal linear weights, while in non-smooth regions the stiff diffusion
degenerates. The algorithm is described in detail, and numerous numerical
experiments are conducted to demonstrate the effectiveness of such a treatment
and the good performance of our method. The stiffness of the nonlinear
parabolic partial differential equations (PDEs) is resolved well, and large
time-step size computations are achieved.
