Fourth- and Higher-order Interface Tracking Via Mapping and Adjusting Regular Semianalytic sets Represented by Cubic Splines
Fourth- and Higher-order Interface Tracking Via Mapping and Adjusting Regular Semianalytic sets Represented by Cubic Splines
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DOI：
10.1137/17m1149328
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发表时间：
2018-11
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影响因子：
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通讯作者：
Qinghai Zhang
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文献类型：
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作者：
Qinghai Zhang
这项工作是我们二维界面跟踪研究路线的进一步发展和顶峰[Zhang and Liu, J. Comput.物理学，27(2008)，第4063--4088页；张，SIAM J.数字。分析，51(2013)，第2822--2850页；张和福格尔森，SIAM J. Sci。计算机，36 (2014)，第 A2369--A2400 页；张和福格尔森，SIAM J. Numer。分析，54(2016)，第530--560页]。之前，我们提出了显式界面跟踪的分析框架和四阶界面跟踪方法。在本文中，我们继续提出一种新方法，称为三次 MARS 方法，其特点是（1）由来自不同数学分支的成熟概念组成的坚实理论基础，（2）三次样条接口的表示，（3）线性/曲线多边形上布尔运算的新颖算法，以及（4）通过对欧拉网格大小 $h$ 和拉格朗日长度尺度之间关系的精细控制来实现四阶、六阶和八阶精度$h_L$ ...
This work is a further development and the culmination along our research line of interface tracking in two dimensions [Zhang and Liu, J. Comput. Phys., 27 (2008), pp. 4063--4088; Zhang, SIAM J. Numer. Anal., 51 (2013), pp. 2822--2850; Zhang and Fogelson, SIAM J. Sci. Comput., 36 (2014), pp. A2369--A2400; Zhang and Fogelson, SIAM J. Numer. Anal., 54 (2016), pp. 530--560]. Previously, we have proposed an analytic framework for explicit interface tracking and a fourth-order interface tracking method. In this paper, we continue our approach to propose a new method, called the cubic MARS method, that features (1) a solid theoretical foundation consisting of well developed concepts from distinct branches of mathematics, (2) a representation of the interface with cubic splines, (3) a novel algorithm for Boolean operations on linear/curvilinear polygons, and (4) fourth-, sixth-, and eighth-order accuracy via a refined control of the relation between the Eulerian grid size $h$ and a Lagrangian length scale $h_L$ ...