  In this paper, we present a novel numerical scheme for solving a class of
nonlinear degenerate parabolic equations with non-smooth solutions. The
proposed method relies on a special kernel based formulation of the solutions
found in our early work on the method of lines transpose and successive
convolution. In such a framework, a high order weighted essentially
non-oscillatory (WENO) methodology and a nonlinear filter are further employed
to avoid spurious oscillations. High order accuracy in time is realized by
using the high order explicit strong-stability-preserving (SSP) Runge-Kutta
method. Moreover, theoretical investigations of the kernel based formulation
combined with an explicit SSP method indicates that the combined scheme is
unconditionally stable and up to third order accuracy. Evaluation of the kernel
based approach is done with a fast $\mathcal{O}(N)$ summation algorithm. The
new method allows for much larger time step evolution compared with other
explicit schemes with the same order accuracy, leading to remarkable
computational savings.
