Combined WENO schemes for increasing the accuracy of the numerical solution of conservation laws

TL;DR

The paper presents WENO-C, a combined-weighted essentially non-oscillatory scheme that harnesses all sub-stencils to raise accuracy without sacrificing non-oscillatory behavior. It constructs intermediate fluxes from same-order sub-stencils and then blends them with total weights guided by total smoothness indicators, with a tunable parameter $p$ to emphasize higher-order stencils. The method extends to WENO-ZC by using a WENO-Z-like definition of weights, and demonstrates improved accuracy across linear advection and Euler test problems, including complex shock interactions, while preserving the ENO property. This approach offers a practical route to higher-order, robust shock-capturing schemes for hyperbolic conservation laws.

Abstract

In this article, we introduce a new method which allows utilizing all the available sub-stencils of a WENO scheme to increase the accuracy of the numerical solution of conservation laws while preserving the non-oscillatory property of the scheme. In this method, near a discontinuity, if there is a smooth sub-stencil with higher-order of accuracy, it is used in the reconstruction procedure. Furthermore, in smooth regions, all the sub-stencils of the same order of accuracy form the stencil with the highest order of accuracy as the conventional WENO scheme. The presented method is assessed using several test cases of the linear wave equation and one- and two-dimensional Euler's equations of gas dynamics. In addition to the original weights of WENO schemes, the WENO-Z approach is used. The results show that the new method increases the accuracy of the results while properly maintaining the ENO property.

Figures (11)

• Figure 1: All sub-stencils inside $S^5_2$ ($k=3$).
• Figure 2: All sub-stencils inside $S^7_3$ ($k=4$).
• Figure 3: The distribution of the weights near a discontinuity; top-left) the function (\ref{['e:wave:init']}), bottom-left) total weights of the third- and fourth-order stencils, top-right) weights of the third-order sub-stencils, bottom-right) weights of the fourth-order sub-stencils.
• Figure 5: Numerical solution of the advection equation by the WENO5-JS and WENO5-C schemes using $N = 201$ at $t = 8$ using different values for the power parameter $p$.
• Figure 7: The zoomed view of Figs. \ref{['f:wave:8:5:JS']} (left) and \ref{['f:wave:8:5:Z']} (right) around the right discontinuity of the square wave.