A Modified Hybrid High-Order Essentially Non-Oscillatory Compact Difference Scheme for Nonlinear Degenerate Parabolic Equations (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, we propose a modified hybrid high-order essentially non-oscillatory compact difference scheme for both one- and two-dimensional nonlinear degenerate parabolic equations that may exist discontinuous solutions. A cell-centered compact scheme is applied to approximate the second-order derivative, where the scalar values on a cell-centered mesh are obtained by a modified hybrid essentially non-oscillatory interpolation formulation, which is constructed by coupling the compact interpolation scheme with an odd-even order weighted essentially non-oscillatory (WENO) scheme. Furthermore, a positivity-preserving limiter is designed to circumvent the numerical solutions of possible non-physical properties. The main purpose of constructing a modified hybrid cell-centered compact scheme is based on two considerations. On the one hand, it helps to avoid loss of accuracy caused by directly incorporating the WENO scheme into the cell-centered compact scheme. On the other hand, it provides a feasible way for the compact difference method to solve discontinuous problems with high-order derivatives. Here, we mainly take the numerical approximation of the second-order derivative as an example for detailed process reasoning. Through numerical tests, high-order, high resolution and essentially non-oscillatory performance of the schemes presented are verified.