In this paper we analyze the weighted essentially non-oscillatory (WENO) schemes in the finite volume framework by examining the first step of the explicit third-order total variation diminishing Runge-Kutta method. The rationale for the improved performance of the finite volume WENO-M, WENO-Z and WENO-ZR schemes over WENO-JS in the first time step is that the nonlinear weights corresponding to large errors are adjusted to increase the accuracy of numerical solutions. Based on this analysis, we propose novel Z-type nonlinear weights of the finite volume WENO scheme for hyperbolic conservation laws. Instead of taking the difference of the smoothness indicators for the global smoothness indicator, we employ the logarithmic function with tuners to ensure that the numerical dissipation is reduced around discontinuities while the essentially non-oscillatory property is preserved. The proposed scheme does not necessitate substantial extra computational expenses. Numerical examples are presented to demonstrate the capability of the proposed WENO scheme in shock capturing.

翻译：本文通过分析显式三阶总变差递减龙格-库塔方法的第一步，在有限体积框架下对加权本质无振荡（WENO）格式进行了研究。有限体积WENO-M、WENO-Z和WENO-ZR格式在第一时间步中相较于WENO-JS格式性能提升的原因在于，其通过调整对应大误差的非线性权重提高了数值解的精度。基于此分析，我们针对双曲守恒律提出了一种新型有限体积WENO格式的Z型非线性权重。该方法不采用光滑度因子差作为全局光滑度指标，而是使用带有调节器的对数函数，确保在间断附近降低数值耗散的同时保持本质无振荡特性。所提格式无需显著增加额外计算成本。数值算例展示了该WENO格式在激波捕捉方面的能力。