Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution
Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution
复制标题
DOI：
10.4208/cicp.150215.250515a
复制
发表时间：
2015-11
影响因子：
3.7
通讯作者：
Qin Li;Pengxin Liu;Hanxin Zhang
中科院分区：
文献类型：
--
作者：
Qin Li;Pengxin Liu;Hanxin Zhang
映射函数的方法首先由Henrick等人[J. Comput. Phys.207：542-547（2005）]对五阶韦诺格式调整[0，1]中的非线性权值，满足了收敛阶数的要求，提高了格式的性能。与Henrick方法不同，本文提出了分段多项式函数的概念，并得到了相应的韦诺格式。新方法的优点是，在线性权值处，函数的轮廓比较平缓（或者映射后的非线性权值与线性权值接近），有利于分辨率的提高。此外，该函数还具有在[0，1]的两个端点附近快速收敛到恒等映射的灵活性，有利于提高数值稳定性。相应地构造了四阶、五阶和六阶多项式函数，在平坦性和收敛性方面各有侧重。其中，五阶版本的外形最为平坦。为了检验方法的性能，我们分别对一维Shu-Osher问题、二维Riemann问题和双马赫反射问题进行了数值实验，并与WENO-M、WENO-Z和WENO-NS方法进行了比较。所提出的新方法显示出最好的分辨率描述剪切层不稳定性的黎曼问题，他们也表明高分辨率的计算双马赫反射，只有这些建议的计划成功地解决了涡配对现象。其他研究表明，单一的多项式映射函数并没有优于建议分段的，它是没有明显的好处，使用所提出的方法对对称五阶韦诺。总体而言，五阶分段多项式和相应的韦诺格式的分辨率提高建议。AMS科目分类：65 M06、65 M12
The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifthorder WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick’s method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifthand sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement. AMS subject classifications: 65M06, 65M12