A Family of Even-Order Central-Upwind WENO Schemes with Averaged Downwind and Novel Global Smoothness Indicators

TL;DR

This paper introduces a new sixth-order central-upwind WENO scheme with an averaged downwind smoothness indicator, achieving high accuracy, efficiency, and sharp resolution of discontinuities for hyperbolic conservation laws.

Contribution

It develops a novel averaging approach for the downwind stencil's smoothness indicator, enhancing accuracy and efficiency without complex tuning.

Findings

WENO-ZA6 achieves optimal convergence rates at critical points.

The scheme provides sharp resolution of discontinuities.

It reduces computational time by approximately 15-21% compared to existing schemes.

Abstract

We propose a simple yet effective local smoothness indicator for the downwind stencil in central-upwind weighted essentially non-oscillatory (WENO) schemes of even order for hyperbolic conservation laws. Starting from an odd-order upwind WENO scheme, we construct an even-number-of-points stencil by incorporating a downwind substencil whose smoothness indicator is the arithmetic mean of all local smoothness indicators. This straightforward averaging approach incorporates regularity information from the entire stencil without requiring additional tuning parameters or complex formulations. Combined with affine-invariant Z-type nonlinear weights and a carefully designed global smoothness indicator, the resulting scheme, termed WENO-ZA6 for the sixth-order case, achieves optimal convergence rates at critical points up to second order, exhibits favorable dispersion and dissipation properties as confirmed by approximate dispersion relation analysis, and provides sharp, essentially non-oscillatory resolution of discontinuities. Numerical experiments on scalar problems and the one- and two-dimensional Euler equations demonstrate that WENO-ZA6 achieves accuracy comparable to or better than existing sixth-order central-upwind schemes (WENO-CU6, WENO-S6) and the seventh-order WENO-Z7, while requiring approximately 15\%--21\% less computational time. The framework extends naturally to fourth-, eighth-, and tenth-order schemes.