Central WENO schemes through a global average weight

TL;DR

This work addresses limitations of classical WENO reconstructions, notably the problematic ideal weights and accuracy loss near smooth extrema, by introducing a central WENO (CWENO) scheme that employs a global average weight computed from Jiang-Shu smoothness indicators. The global weight, defined as $\omega_j=\frac{(r+1)^2}{(\sum_{i=0}^r(I_i+\varepsilon)^m)(\sum_{i=0}^r 1/(I_i+\varepsilon)^m)}$, governs a blend between the full-stencil reconstruction and sub-stencil reconstructions, while subweights are formed from $\alpha_{i,j}=c_i/(I_{i,j}+\varepsilon)^s$. The authors provide a detailed accuracy analysis showing how to choose $m$, $s$, and $\varepsilon$ to recover the optimal order $2r+1$ on smooth data and maintain stability across discontinuities, and they validate the method with extensive numerical experiments. Comparisons with WENO and CWENO-LPR demonstrate improved resolution, robustness to smooth extrema, and favorable computational cost, including non-centered stencil scenarios. The results indicate that CWENO with a global average weight offers a practical, high-order, non-oscillatory reconstruction framework for hyperbolic conservation laws, with potential extensions to unstructured grids and boundary extrapolation in future work.

Abstract

A novel central weighted essentially non-oscillatory (central WENO; CWENO)-type scheme for the construction of high-resolution approximations to discontinuous solutions to hyperbolic systems of conservation laws is presented. This procedure is based on the construction of a global average weight using the whole set of Jiang-Shu smoothness indicators associated to every candidate stencil. By this device one does not to have to rely on ideal weights, which, under certain stencil arrangements and interpolating point locations, do not define a convex combination of the lower-degree interpolating polynomials of the corresponding sub-stencils. Moreover, this procedure also prevents some cases of accuracy loss near smooth extrema that are experienced by classical WENO and CWENO schemes. These properties result in a more flexible scheme that overcomes these issues, at the cost of only a few additional computations with respect to classical WENO schemes and with a smaller cost than classical CWENO schemes. Numerical examples illustrate that the proposed CWENO schemes outperform both the traditional WENO and the original CWENO schemes.

Figures (4)

• Figure 1: Example 5: density field, discretization with $200$ points, $s_2=2$, $T=1.8$.
• Figure 2: Example 5: efficiency plot.
• Figure 3: Example 6: enlarged view of turbulent zone (Schlieren plot).
• Figure 4: Example 7: enlarged view of turbulent zone (Schlieren plot).