Hybrid HWENO Method for Nonlinear Degenerate Parabolic Equations

In this paper, a hybrid Hermite weighted essentially non-oscillatory scheme is proposed for nonlinear degenerate parabolic equations which may contain discontinuity in the solution. The present scheme is constructed by applying the hybrid HWENO method based on the zero- and first-order moment with DDG flux to discrete the diffusion term in spatial direction and the third-order TVD Runge-Kutta method in temporal direction. A troubled-cell indicator is first used to identify the cells in which the discontinuity may exist, then the first-order moment in the troubled-cell is reconstructed by fifth-order HWENO scheme. To avoid spurious oscillation, the HWENO reconstruction is performed when the reconstruction stencils contain troubled-cell, otherwise linear reconstruction is performed straightforwardly. Compared with WENO schemes, the present scheme has advantages: (1) compactness, only immediate neighbor cells are used in the reconstruction procedure; (2) accuracy, the numerical errors by the present scheme are smaller than those by WENO schemes. Some benchmarks for one- and two-dimensional parabolic equations to demonstrate the high order accuracy and non-oscillatory performance of the present scheme.