A Family of Even-Order Central-Upwind WENO Schemes with Averaged Downwind and Novel Global Smoothness Indicators
This work addresses computational efficiency and accuracy in numerical simulations for hyperbolic conservation laws, offering an incremental improvement over prior WENO schemes.
The authors tackled the problem of improving even-order central-upwind WENO schemes for hyperbolic conservation laws by proposing a novel local smoothness indicator based on averaging downwind stencils, resulting in the WENO-ZA6 scheme that achieves optimal convergence rates, sharp discontinuity resolution, and reduces computational time by 15%–21% compared to existing methods.
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.