A sixth-order central WENO scheme for nonlinear degenerate parabolic equations

TL;DR

This work develops a sixth-order finite-difference central WENO (CWENO) scheme with Z-type nonlinear weights to solve nonlinear degenerate parabolic equations, notably the porous medium equation $u_t=(u^m)_{xx}$ with $m>1$. By introducing a centered polynomial, the method ensures positive linear weights and forms a central flux $\hat{g}^ ext{C}_{i+1/2}$ alongside a global smoothness indicator $\tau_6$, yielding nonlinear weights $\omega^{\text{CWENO}}_k$ that achieve sixth-order accuracy in smooth regions. The paper derives sufficient conditions for accuracy, analyzes the impact of inflection points, and demonstrates, via extensive 1D and 2D numerical experiments, that CWENO-DZ delivers high accuracy with reduced oscillations compared to WENO-LSZ and MWENO. The approach is computationally efficient and extendable to convection–diffusion and higher dimensions, offering a robust tool for resolving sharp interfaces and free boundaries in nonlinear degenerate parabolic problems.

Abstract

In this paper we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction in order to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $L^2$-norm smoothness indicators, yielding the new WENO scheme with more accurate resolution. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.

Figures (14)

• Figure 1: The numerical flux $\hat{g}_{i+1/2}$ is constructed on the stencil $S^6 = \{ x_{i-2}, \cdots, x_{i+3} \}$ with six uniform points, as well as three 4-point substencils $S_0, S_1, S_2$.
• Figure 2: Barenblatt solution profiles for Example \ref{['ex:Barenblatt']} with $m=5$ at $T=2$ (left), close-up view of the solutions in the boxes on the left/right (middle/right) computed by WENO-LSZ (red), MWENO (green) and CWENO-DZ (blue) with $N=160$. The dashed black lines are the exact solution.
• Figure 3: Barenblatt solution profiles for Example \ref{['ex:Barenblatt']} with $m=7$ at $T=2$ (left), close-up view of the solutions in the boxes on the left/right (middle/right) computed by WENO-LSZ (red), MWENO (green) and CWENO-DZ (blue) with $N=160$. The dashed black lines are the exact solution.
• Figure 4: Barenblatt solution profiles for Example \ref{['ex:Barenblatt']} with $m=9$ at $T=2$ (left), close-up view of the solutions in the boxes on the left/right (middle/right) computed by WENO-LSZ (red), MWENO (green) and CWENO-DZ (blue) with $N=160$. The dashed black lines are the exact solution.
• Figure 5: Solution profiles for PME \ref{['eq:pme']} ($m=5$) with the initial condition \ref{['eq:two_box_init_sh']} at $t = 0.5$ (left), $1.0$ (middle) and $1.5$ (right) approximated by WENO-LSZ (red), MWENO (green) and CWENO-DZ (blue) with $N = 220$. The black lines are generated by MWENO with $N = 11000$.

Theorems & Definitions (10)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7
• Example 4.8
• Example 4.9
• Example 4.10