An efficient sixth-order finite difference WENO scheme for nonlinear degenerate parabolic equations

In this paper, we propose an efficient sixth-order finite difference weighted essentially nonoscillatory (WENO) scheme for numerically solving nonlinear degenerate parabolic (NDP) equations which may contain discontinuous solutions. For the discretization of second-order spatial derivative terms, a new sixth-order WENO scheme is directly implemented. This new scheme is constructed by a convex combination of a fourth-degree polynomial with three quadratic polynomials in a traditional fashion. To improve the scheme’s efficiency, a new smoothness indicator for the big stencil is carefully designed. This scheme has three properties: (1) it achieves sixth-order accuracy in smooth areas and third-order accuracy near singularities; (2) the linear weights can be any artificial positive numbers with the symmetry requirement and that their sum equals one; (3) it is efficient since no mapping procedure and negative weights are involved. Numerical tests in one- and two-dimensional cases are presented to illustrate that the new WENO scheme provides similar numerical results with smaller computational costs in comparison to the WENO schemes developed by Liu et al. [SIAM J. Sci. Comput. 33, 939 (2011)] and Jiang [J. Sci. Comput. 86, 16 (2021)].