• 论文 • 上一篇    下一篇

求解双曲守恒律的紧-WENO杂交格式及RBF-FD间断检测方法

杨洋, 曾维新, 高振, 王保山   

  1. 中国海洋大学数学科学学院, 青岛 266100
  • 收稿日期:2019-09-10 出版日期:2020-09-15 发布日期:2020-09-15
  • 基金资助:

    国家自然科学基金(11871443)资助.

杨洋, 曾维新, 高振, 王保山. 求解双曲守恒律的紧-WENO杂交格式及RBF-FD间断检测方法[J]. 数值计算与计算机应用, 2020, 41(3): 232-245.

Yang Yang, Don Wai Sun, Gao Zhen, Wang Bao-Shan. HYBRID COMPACT-WENO SCHEME WITH RBF-FD BASED DISCONTINUITY DETECTION METHOD FOR HYPERBOLIC CONSERVATION LAWS[J]. Journal of Numerical Methods and Computer Applications, 2020, 41(3): 232-245.

HYBRID COMPACT-WENO SCHEME WITH RBF-FD BASED DISCONTINUITY DETECTION METHOD FOR HYPERBOLIC CONSERVATION LAWS

Yang Yang, Don Wai Sun, Gao Zhen, Wang Bao-Shan   

  1. School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
  • Received:2019-09-10 Online:2020-09-15 Published:2020-09-15
设计准确、鲁棒和高效的间断检测方法,在任意给定的网格单元和任意时间上衡量解的光滑度,是杂交格式的关键问题.本文从有限差分形式的径向基函数(RBF-FD)方法出发设计间断检测方法,为解决其阈值参数的选取问题,采用数据分析中Tukey's箱线图方法增强算法的鲁棒性,最终提出一种准确、鲁棒和高效的RBF-FD间断检测方法.数值结果分析表明,RBF-FD间断检测方法在间断捕捉的准确性、鲁棒性和效率等方面优于已有的RBF方法和单调多项式插值方法,并且与五阶WENO-Z格式相比,应用RBF-FD方法的杂交格式可以加速2倍左右.
An accurate, robust and efficient discontinuity detection method to detect the smoothness of solutions at any given grid point and time, is the key problem of hybrid schemes. By using finite difference radial basis function (RBF-FD) method and increasing the robustness of the discontinuity detection method by employing the Tukey's boxplot method in data analysis, an accurate, robust and efficient RBF-FD based discontinuity detection method is proposed. Several classical benchmark examples show that the RBF-FD method captures shocks better than the original RBF discontinuity detection method and the monotone interpolation method in terms of accuracy, robustness, and efficiency. The hybrid scheme with the proposed RBF-FD method is about two times more efficient in the CPU times than the fifth order WENO-Z scheme.

MR(2010)主题分类: 

()
[1] Jiang G S, Shu C W, Efficient implementation of weighted ENO Schemes[J]. J. Comput. Phys., 1996, 126(1):202-228.

[2] Borges R, Carmona M, Costa B, Don W S, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. J. Comput. Phys., 2008, 227(6):3191-3211.

[3] Castro M, Costa B, Don W S, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws[J]. J. Comput. Phys., 2011, 230(5):1766-1792.

[4] Cockburn B, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II:General framework[J]. Math. Comput., 1989, 52:411-435.

[5] Krivodonova L, Xin J, Remacle J F, Chevaugeon N, Flaherty J. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws[J]. Appl. Numer. Math., 2004, 48:323-338.

[6] Costa B, Don W S. High order Hybrid Central-WENO finite difference scheme for conservation laws[J]. J. Comput. Appl. Math., 2007, 204(2):209-218.

[7] Gao Z, Don W S. Mapped Hybrid Central-WENO Finite Difference Scheme for Detonation Waves Simulations[J]. J. Sci. Comput., 2013, 55:351-371.

[8] Costa B, Don W S. Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws[J]. J. Comput. Phys., 2007, 224:970-991.

[9] Gao Z, Wen X, Don W S. Enhanced robustness of the hybrid Compact-WENO finite difference scheme for hyperbolic conservation laws with multi-resolution analysis and Tukey's boxplot method[J]. J. Sci. Comput., 2017, 73:736-752.

[10] Don W S, Gao Z, Li P, Wen X. Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws[J]. SIAM J. Sci. Comput., 2016, 38(2):A691-A711.

[11] Guo J, Jung J H. A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method[J]. J. Appl. Num. Math., 2017, 112:27-50.

[12] Wang B S, Don W S, Gao Z, Wang Y, Wen X. Hybrid Compact-WENO Finite Difference Scheme with Radial Basis Function Based Shock Detection Method for Hyperbolic Conservation Laws[J]. SIAM J. Sci. Comput., 2018, 40(6):A3699-A3714.

[13] 王保山. 基于径向基函数的间断检测算法及其在高精度杂交格式中的应用[D]. 青岛:中国海洋大学, 2018.

[14] Feng R, Duan J. High Accurate Finite Differences Based on RBF Interpolation and its Application in Solving Differential Equations[J]. J Sci Comput., 2018, 76:1785-1812.

[15] Tukey J W. Exploratory data analysis, 1st ed.[M]. Reading:Addison-Wesley, 1977.

[16] Vuik M J, Ryan J K. Automated parameters for troubled-cell indicators using outlier detection[J]. SIAM J. Sci. Comput., 2016, 38(1):A84-A104.

[17] Vasilyev O, Lund T, Moin P. A general class of commutative filters for LES in complex geometries[J]. J. Comput. Phys., 1998, 146(1):82-104.

[18] Buhmann M D. Radial Basis Functions:Theory and Implementations[M]. Cambridge:Cambridge University Press, 2003.

[19] Fornberg B, Wright G, Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions[J]. Comput. Math. Appl., 2004, 47:37-55.

[20] Woodward P, Colella P. The numerical simulation of two dimensional fluid flow with strong shocks[J]. J. Comput. Phys., 1984, 54:115-173.

[21] Lax P D, Liu X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. SIAM J. Sci. Comput., 1998, 19:319-340.

[22] Yee H C, Sandham N D, Djomehri M J. Low dissipative high order shock-capturing methods using characteristic-based filters[J]. J. Comput. Phys., 1999, 150:199-238.
[1] 徐捷, 高振, 曾维新, 王保山. 高阶驻点上精度保持的六阶WENO有限差分格式[J]. 数值计算与计算机应用, 2020, 41(1): 68-82.
[2] 张学莹,赵宁. 求解多介质流体界面不稳定性问题的高精度WENO数值模拟方法[J]. 数值计算与计算机应用, 2007, 28(2): 81-90.
阅读次数
全文


摘要