In this paper, a compact Hermite weighted essentially non-oscillatory (CHWENO) scheme is proposed for nonlinear degenerate parabolic equations. The scheme is developed by integrating a compact central difference discretization with a nonlinear Hermite WENO (HWENO) methodology. Compared with the HWENO scheme presented in [3], the CHWENO scheme has two significant advantages: (1) High efficiency—Compared with the reference HWENO scheme, the proposed CHWENO scheme has the significant improvement because the derivative values of the solutions are obtained directly using a compact central difference scheme, and we do not need to solve the auxiliary derivative equations. Numerical results show that CHWENO schemes reduce CPU time by approximately 30% for one-dimensional problems and by up to approximately 55% for two-dimensional cases. (2) Simplicity—The proposed method is simpler and more straightforward to implement for two-dimensional problems, since it can be applied dimension by dimension. Numerical experiments on the porous medium equation and the Buckley–Leverett equation confirm that the proposed scheme can effectively suppress oscillations in the presence of large gradients and maintain nonlinear stability.
