High order implicit finite difference schemes with a semi-implicit WENO reconstruction for nonlinear degenerate parabolic equations

Abstract

In this paper, we develop implicit finite difference schemes with a semi-implicit weighted essentially non-oscillatory (WENO) reconstruction for nonlinear degenerate parabolic equations. Such degenerate equations would exist finite speed front propagation which is similar to discontinuous solutions for hyperbolic equations. We propose to use the finite difference WENO scheme with an implicit time discretization, to avoid the parabolic time step condition from an explicit time discretization, namely Δt=O(Δx2) where Δx is the mesh size. We further simplify the resulting nonlinear system by employing a semi-implicit WENO reconstruction, that is, nonlinear weights are treated explicitly while keeping linear reconstructions of derivatives implicitly. The novelty is that the semi-implicit WENO reconstruction maintains the original high order accuracy both in space and in time, as well as unconditional stability as a fully implicit scheme, but the scheme is much simpler in complexity than the fully implicit one. Numerical experiments are performed to show the high order accuracy, large time step conditions with better efficiency as compared to an explicit scheme, and good performances for capturing front propagation.