On the efficient computation of smoothness indicators for a class of WENO reconstructions

TL;DR

The paper tackles the expensive computation of smoothness indicators in WENO reconstructions for hyperbolic conservation laws by introducing a new set of smoothness indicators with linear cost in the order, enabling fast WENO (FWENO) schemes within Yamaleev–Carpenter-type reconstructions. The core idea is to replace the quadratic Jiang–Shu indicators with $I_{r,i} = \sum_{j=1}^{r-1} (f_{-r+i+j+1}-f_{-r+i+j})^2$ and to compute them via a simple recurrence, preserving the non-linear weight properties and accuracy. Theoretical results show that, for smooth data, $I_{r,i} = \bar O(h^{2n+2})$, while near discontinuities certain $I_{r,i}$ remain $O(1)$ or drop to $O(h^2)$, ensuring the expected WENO accuracy $q_r(x_{1/2}) = f(x_{1/2}) + O(h^{2r-1})$ except in a known edge case $n=2r-3$ where the order may drop to $O(h^{2r-2})$. Numerical experiments in 1D and 2D hyperbolic problems show FWENO achieves comparable or better accuracy than JS-WENO and YC-WENO while substantially reducing computational cost, demonstrating practical impact for high-order schemes on uniform grids and suggesting benefits for nonuniform grids and boundary extrapolation in future work.

Abstract

Common smoothness indicators used in Weighted Essentially Non\--Os\-cil\-la\-to\-ry (WENO) reconstructions [Jiang, G.S., Shu, C.W.: Efficient implementation of {Weighted} {ENO} schemes, J.\ Comput.\ Phys. \textbf{126}, 202--228 (1996)] have quadratic cost with respect to the order. A set of novel smoothness indicators with linear cost of computation with respect to the order is presented. These smoothness indicators can be used in the context of schemes of the type introduced by Yamaleev and Carpenter [Yamaleev, N.K., Carpenter, M.H.: A systematic methodology to for constructing high-order energy stable WENO schemes. J. Comput. Phys. \textbf{228}(11), 4248-4272 (2009)]. The accuracy properties of the resulting non-linear weights are the same as those arising from using the traditional Jiang-Shu smoothness indicators in Yamaleev-Carpenter-type reconstructions. The increase of the efficiency and ease of implementation are shown.

Figures (12)

• Figure 1: Theoretical cost comparison between WENO interpolators.
• Figure 2: Example 2 (inviscid Burgers equation): numerical solution for $N=80$ (top) and enlarged views (bottom left and right) at $T=12$ produced by fifth-order schemes with conventional (JS-WENO5 and YC-WENO5) and new (FWENO5) smoothness indicators. The reference solution with $N=16000$ cells is also shown.
• Figure 3: Example 3 (Shu-Osher problem, 1D Euler equations of gas dynamics): numerical solution for $N=200$ (top left) and enlarged views (top right, bottom left and bottom right) at $T=1.8$ produced by fifth-order schemes with conventional (JS-WENO5 and YC-WENO5) and new (FWENO5) smoothness indicators. The reference solution with $N=16000$ cells is also shown.
• Figure 4: Example 3 (Shu-Osher problem, 1D Euler equations of gas dynamics): numerical solution for $N=400$ (top left) and enlarged views (top right, bottom left and bottom right) at $T=1.8$ produced by fifth-order schemes with conventional (JS-WENO5 and YC-WENO5) and new (FWENO5) smoothness indicators. The reference solution with $N=16000$ cells is also shown.
• Figure 5: Example 3 (Shu-Osher problem, 1D Euler equations of gas dynamics): efficiency plot for fifth-order schemes, corresponding to the numerical solution at $T=1.8$.

Theorems & Definitions (11)

• lemma 1
• proof
• lemma 2
• proof
• remark 1
• lemma 3
• proof
• remark 2
• theorem 1
• proof