High-order WENO discretization of degenerate parabolic problems utilizing arc-length smoothness quantification

Purpose This study aims to propose a sixth-order finite difference scheme based on weighted essentially nonoscillatory (WENO) reconstruction to simulate nonlinear degenerate parabolic (NDP) equations. Design/methodology/approach With the aim of constructing this scheme, a novel procedure based on the length of a curve is provided for defining the smoothness parameter in the nonlinear weights of the new WENO scheme. Findings This new WENO scheme has two major noteworthy aspects. First, it is designed to be less oscillatory, which is crucial when dealing with complex solutions that may exhibit oscillatory behavior. Second, it continues to achieve the designated accuracy order even under the challenge of critical points in the exact solution. The assertion is substantiated through theoretical evidence and computational results, which are presented using different norms. Originality/value By developing advanced nonlinear weights, the paper offers a novel WENO technique tailored to multidimensional NDP equations.