JS-type and Z-type weights for fourth-order central-upwind weighted essentially non-oscillatory schemes

TL;DR

The paper addresses the challenge of designing a high-order central-upwind WENO scheme for hyperbolic conservation laws by introducing a downwind substencil with an averaged smoothness indicator. It develops JS-type and Z-type nonlinear weights for a fourth-order scheme (WENO4-JS and WENO4-ZA), establishing a Fourth-Order accuracy condition $\omega_k-d_k = O(\Delta x^3)$ and showing that Z-type weights can satisfy this under appropriate parameter choices. Through detailed analysis and extensive 1D/2D numerical experiments on linear advection and Euler equations, the study demonstrates that WENO4-ZA delivers fourth-order accuracy in smooth regions and sharper resolution near discontinuities due to its closer alignment to linear weights. The results indicate that the central-upwind WENO framework with ZA weighting offers robust, high-resolution performance for complex compressible flows, with potential advantages over traditional WENO schemes in dissipation control and shock-capturing fidelity.

Abstract

The central-upwind weighted essentially non-oscillatory (WENO) scheme introduces the downwind substencil to reconstruct the numerical flux, where the smoothness indicator for the downwind substencil is of critical importance in maintaining high order in smooth regions and preserving the essentially nonoscillatory behavior in shock capturing. In this study, we design the smoothness indicator for the downwind substencil by simply summing up all local smoothness indicators and taking the average, which includes the regularity information of the whole stencil. Accordingly the JS-type and Z-type nonlinear weights, based on simple local smoothness indicators, are developed for the fourth-order central-upwind WENO scheme. The accuracy, robustness, and high-resolution properties of our proposed schemes are demonstrated in a variety of one- and two-dimensional problems.

Figures (9)

• Figure 1: The construction of numerical flux $\hat{f}^-_{i+1/2}$ depends on the stencils $S^3$ and $S^4$ for the respective third- and fourth-order accuracy, as well as three $2$-point substencils $S_0,\: S_1,\: S_2$ for second-order accuracy.
• Figure 2: Density profiles for the Sod's problem \ref{['eq:euler_1d']} and \ref{['eq:sod']} at $T=2$ (top left), close-up view of the solutions in the boxes from left to right (top right, bottom left, bottom right) approximated by WENO3-Z (green), WENO4-JS (blue), WENO4-ZA (red) and WENO5-JS (purple) with $N = 200$. The dashed black lines are the exact solution.
• Figure 3: Density profiles for the Lax's problem \ref{['eq:euler_1d']} and \ref{['eq:lax']} at $T=1.3$ (top left), close-up view of the solutions in the boxes from left to right (top right, bottom left, bottom right) solved by WENO3-Z (green), WENO4-JS (blue), WENO4-ZA (red) and WENO5-JS (purple) with $N = 200$. The dashed black lines are the exact solution.
• Figure 4: Solution profiles for Example \ref{['ex:shock_entropy_wave']} with $k=5$ at $T=2$ (left), close-up view of the solutions in the box (right) computed by WENO3-Z (green), WENO4-JS (blue), WENO4-ZA (red) and WENO5-JS (purple) with $N=400$. The dashed black lines are generated by WENO5-M with $N=4000$.
• Figure 5: Solution profiles for Example \ref{['ex:shock_entropy_wave']} with $k=10$ at $T=2$ (left), close-up view of the solutions in the box (right) computed by WENO3-Z (green), WENO4-JS (blue), WENO4-ZA (red) and WENO5-JS (purple) with $N=800$. The dashed black lines are generated by WENO5-M with $N=8000$.

Theorems & Definitions (7)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7