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：本文通过考察显式三阶总变差递减龙格-库塔方法的第一步，在有限体积框架下分析了加权本质无震荡（WENO）格式。研究发现，有限体积WENO-M、WENO-Z和WENO-ZR格式在第一时间步上优于WENO-JS格式的原因在于：对应于大误差的非线性权重被调整以提高数值解的精度。基于此分析，我们针对双曲守恒律提出了有限体积WENO格式的新型Z型非线性权重。该方法并非采用光滑度指标之差作为全局光滑度指标，而是引入带调节参数的对数函数，确保在不连续区域降低数值耗散的同时保持本质无震荡特性。所提格式无需大量额外计算开销。数值算例展示了该WENO格式在激波捕捉方面的能力。