Abstract The hybridization of the fourth-order central scheme and a third-order WENO (weighted essentially non-oscillatory) scheme aims to obtain fourth-order accuracy in smooth regions and keep ENO property near shock waves. However, according to our numerical tests, such existing hybrid schemes may generate spurious waves near shocks. To overcome this problem, in this paper, a new two-step method is proposed for constructing a fourth-order hybrid central WENO scheme. The first step is using the traditional weighting method to get the third-order WENO scheme. The second step is to get a hybrid scheme by weighting the third-order upwind scheme and the fourth-order central scheme. Then, the final hybrid WENO scheme is obtained by combining the two weighted schemes through a discontinuity-detecting method for a four-point stencil. Various one- and two-dimensional numerical tests are presented to show that the new scheme can avoid the generation of spurious waves near shock waves, keep the ENO property, and get almost fourth-order accuracy in smooth regions including critical points.

中文翻译：

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构造四阶混合中心WENO方案的两步加权法

摘要 四阶中心方案和三阶WENO（加权基本非振荡）方案的混合旨在在光滑区域获得四阶精度并保持ENO特性接近冲击波。然而，根据我们的数值测试，这种现有的混合方案可能会在冲击附近产生杂散波。为了克服这个问题，在本文中，提出了一种新的两步方法来构造四阶混合中心 WENO 方案。第一步是使用传统的加权方法得到三阶WENO方案。第二步是通过对三阶迎风方案和四阶中心方案进行加权得到混合方案。然后，通过四点模板的不连续检测方法将两种加权方案组合在一起，得到最终的混合 WENO 方案。