High-order weighted compact nonlinear scheme for solving degenerate parabolic equations

Solutions of the nonlinear degenerate parabolic equations may be discontinuous. The classical numerical schemes cannot well handle such discontinuities. In this paper, a new sixth-order weighted compact nonlinear scheme for this type of equations is presented. This presented scheme consists of a sixth-order spatially implicit (or compact) or explicit central differencing for the second spatial derivatives at grid nodes and a weighted nonlinear interpolation technique for the first spatial derivatives at cell centers to reduce spurious oscillations near shock waves and discontinuities. Stability analysis of the sixth-order explicit weighted compact nonlinear scheme along with a total variation diminishing Runge–Kutta method for time discretization is given. Numerical results show that the presented scheme can attain the sixth-order accuracy in smooth regions and the non-oscillatory property in capturing discontinuities.