High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws
High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws
复制标题
DOI：
10.1016/j.jcp.2010.11.028
复制
发表时间：
2011-03
期刊：
影响因子：
--
通讯作者：
Marcos Castro;B. Costa;W. Don
中科院分区：
文献类型：
--
作者：
Marcos Castro;B. Costa;W. Don
在[10]中，作者通过添加一个高阶平滑指标设计了一种新的五阶 WENO 有限差分格式，该指标是作为现有低阶平滑指标的简单且廉价的线性组合而获得的。此外，这种被称为 WENO-Z 的新方案的 CPU 成本相当于经典 WENO-JS [2] 的 CPU 成本，并且小于映射的 WENO-M [5]，因为它不涉及非线性权重的映射。在本文中，我们仔细研究了 WENO 子模板的拉格朗日多项式的泰勒展开式以及经典低阶平滑度指标的相关继承对称性，以获得高阶平滑度指标的通用公式，该公式允许将 WENO-Z 方案扩展到所有（奇数）阶精度。我们进一步研究了 WENO-Z 方案在平滑解关键点处的精度提高以及由于新的非线性权重组而导致的独特数值特征，并且我们表明，在数值耗散方面，WENO-Z 占据了 WENO-JS 和 WENO-M 之间的中间位置。给出了一些标准数值实验，例如欧拉方程的一维黎曼初值问题和马赫3激波密度-波相互作用以及二维双马赫激波反射问题。
In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.