• 论文 •

### 求解散射问题快速多极算法截断误差的一种新的估计

1. 西北大学数学学院, 西安 710127
• 收稿日期:2021-07-14 发布日期:2022-12-08
• 基金资助:
国家自然科学基金(11201373)资助.

Li Ruirong, Meng Wenhui. A NEW ESTIMATION OF THE TRUNCATION ERRORS IN THE FAST MULTIPOLE METHOD FOR SCATTERING PROBLEMS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(4): 415-424.

### A NEW ESTIMATION OF THE TRUNCATION ERRORS IN THE FAST MULTIPOLE METHOD FOR SCATTERING PROBLEMS

Li Ruirong, Meng Wenhui

1. School of Mathematics, Northwest University, Xi'an 710127, China
• Received:2021-07-14 Published:2022-12-08

The fast multipole method(FMM) is an efficient and fast algorithm for large-scale multi-particle systems. In the FMM for solving scattering problems, the related expansion and transformation expressions are based on Graf's addition theorem for Bessel functions. The error of the algorithm in practical calculation is introduced by truncating Graf's addition theorem. In this paper, a new estimate for the truncation errors of Graf's addition theorem is given according to characteristics of errors of the FMM. The results have simpler forms and better approximation effects than existing results, which makes them can be better applied to the FMM for solving scattering problems. Numerical experiments verify the effectiveness and accuracy of our results.

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 [1] Rokhlin V. Rapid solution of integral equations of classical potential theory[J]. J. Comput. Phys., 1985, 60:187-207.[2] Rokhlin V. Rapid solution of integral equations of scattering theory in two dimensions[J]. J. Comput. Phys., 1990, 86:414-439.[3] Amini S, Profit A T J. Multi-level fast multipole solution of the scattering problem[J]. Eng. Anal. Bound. Elem., 2003, 27(5):547-564.[4] Rahola J. Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems[J]. BIT. Numer. Math., 1996, 36(2):333-358.[5] Nishimura N. Fast multipole accelerated boundary integral equation methods[J]. Appl. Mech. Rev., 2002, 55(4):299-324.[6] Cheng H, Huang J, Leiterman T J. An adaptive fast solver for the modified Helmholtz equation in two dimensions[J]. J. Comput. Phys., 2006, 211:616-637.[7] Liu Y. Fast multipole boundary element method-theory and applications in engineering[M]. Cambridge University Press, 2009.[8] Colton D L, Kress R. Integral Equation Methods in Scattering Theory[M]. Wiley, New York, 1983.[9] Bowman F. Introduction to Bessel Functions[M]. Dover, New York, 1958.[10] Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[M]. Dover, New York, 1972.[11] Olver F W J. NIST Handbook of Mathematical Functions[M]. Cambridge University Press, 2010.[12] Amini S, Profit A. Analysis of the truncation errors in the fast multipole method for scattering problems[J]. J. Comput. Appl. Math., 2000, 115:23-33.[13] Meng W, Wang L. Bounds for truncation errors of Graf's and Neumann's addition theorems[J]. Numer. Algorithms., 2016, 72(1):91-106.
 [1] 袁龙, 胡齐芽. 复波数Helmholtz方程和时谐Maxwell方程组的平面波间断Petrov-Galerkin方法[J]. 数值计算与计算机应用, 2015, 36(3): 185-196. [2] 柯日焕, 黎稳. 用CCD法离散求解二维Helmholtz方程的数值方法[J]. 数值计算与计算机应用, 2013, 34(3): 221-230. [3] 段艳婷, 王连堂, 徐建丽. 二维Helmholtz方程外问题的数值解法[J]. 数值计算与计算机应用, 2011, 32(1): 57-63. [4] 孟文辉, 崔俊芝. 求解二维随机多区域声波散射问题的快速多极边界元方法[J]. 数值计算与计算机应用, 2010, 31(2): 141-152. [5] 黄振全,陈志武,何云. 低阶实时最优Runge-Kutta算法[J]. 数值计算与计算机应用, 2008, 29(2): 81-88.