High Order Finite Difference WENO Schemes for Nonlinear Degenerate Parabolic Equations

High order accurate weighted essentially non-oscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous solutions. A porous medium equation is used as an example to demonstrate the algorithm structure and performance. Two different formulations of WENO schemes, one approximating directly the second derivative term using a conservative flux difference, and another approximating this term by first rewriting it as two first derivative terms using an auxiliary variable before applying the WENO procedure on those first derivatives, are discussed and compared. Numerical examples are provided to demonstrate the accuracy and non-oscillatory performance of these schemes.