高振荡Hankel核积分方程的高效数值算法

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 611-619.

高振荡Hankel核积分方程的高效数值算法

吴清华^1,²()

¹ 湘潭大学数学与计算科学学院湖南湘潭 411105
² 湖南科技学院理学院湖南永州 425199

收稿日期:2017-03-16 出版日期:2019-06-26 发布日期:2019-06-27
作者简介:吴清华, jackwqh@163.com
基金资助:
国家自然科学基金(11701170);湖南省自然科学基金(2017JJ3029);湖南省青年骨干教师项目

Efficient Numerical Methods for Integral Equations with Oscillatory Hankel Kernels

Qinghua Wu^1,²()

¹ School of Mathematics and Computational Science, Xiangtan University, Hunan Xiangtan 411105
² College of Science, Hunan University of Science and Engineering, Hunan Yongzhou 425199

Received:2017-03-16 Online:2019-06-26 Published:2019-06-27
Supported by:
the NSFC(11701170);the NSF of Hunan Province(2017JJ3029);the Young Core Teacher Foundation of Hunan Province

摘要/Abstract

摘要：

该文研究入射波为时谐波的情形下，二维声散射问题中的一类边界积分方程（BIE）的数值解法.快速多极方法（FMM）是一个当下流行且非常高效的求解这类积分方程的方法.但是当快速多极方法直接应用于高频声散射问题时，会产生高振荡积分的计算问题.经典的数值积分方法计算这些高振荡积分非常困难并且随着频率的增加计算代价快速增加.因此，该文考虑将快速多极方法和高振荡积分方法相结合提出一种求解带高振荡Hankel核的边界积分方程的数值方法.首先应用边界元方法（BEM）离散积分方程，用快速多极方法加速求解；其次对所涉及的高振荡积分将通过Clenshaw-Curtis Filon（CCF）进行高效计算；最后通过数值算例检验方法的有效性和精确性.

关键词: 高振荡Hankel核, 高振荡积分方程, 快速多极方法

Abstract:

In this paper, we consider the numerical solution of boundary integral equations (BIE) arise in the study of the 2D scattering of a time-harmonic acoustic incident plane wave. Fast multipole method (FMM) is a very efficient and popular algorithm for the rapid solution of boundary value problems. However, when the FMM method is used for high frequency acoustic wave problems, it will give rise to the computation of oscillatory integrals. The standard quadrature methods are exceedingly difficult to calculate these oscillatory integrals and the computation cost steeply increases with the frequency. We apply the boundary element method (BEM) to discretize the BIE and use the FMM to accelerate the solutions of BEM. Oscillatory integrals are calculated by using efficient Clenshaw-Curtis Filon (CCF) methods. The effectiveness and accuracy of the proposed method are tested by numerical examples.

Key words: Oscillatory hankel kernels, Highly oscillatory integral equations, Fast multipole method

中图分类号:

O241.83

吴清华. 高振荡Hankel核积分方程的高效数值算法[J]. 数学物理学报, 2019, 39(3): 611-619.

Qinghua Wu. Efficient Numerical Methods for Integral Equations with Oscillatory Hankel Kernels[J]. Acta mathematica scientia,Series A, 2019, 39(3): 611-619.

图/表 3

表 1

表 2

表 3

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$\omega$	$m$	$I_m[f]$	$\|I1[f]-I_m[f]\|$
$10^3$	$4$	$-0.012459303385353 + 0.020469089205557$i	$ 9.5575\times 10^{-6}$
	$8$	$-0.012451205919241 + 0.020474396104675$i	$ 1.6932\times 10^{-7}$
	$16$	$-0.012451244769401 + 0.020474243043113$i	$1.1403\times 10^{-8}$
	$I_1[f]$	$-0.012451247517459 + 0.020474231975995$i
$10^4$	$4$	$0.000616477977126 + 0.007687006555462$i	$ 5.9071\times 10^{-6}$
	$8$	$0.000619168389738 + 0.007681855738321$i	$ 2.2109\times 10^{-7}$
	$16$	$0.000619041935736 + 0.007681688151854$i	$1.1399\times 10^{-8}$
	$I_1[f]$	$0.000619033274177 + 0.007681680740797$i

$\omega$	$m$	$I_m[f]$	$\|I_2[f]-I_m[f]\|$
$10^3$	$4$	$0.039908808496618 + 0.010081070384206$i	$ 3.8387\times 10^{-5}$
	$8$	$0.039933348663276 + 0.010051682403059$i	$ 4.0079\times 10^{-7}$
	$16$	$0.039933126483704 + 0.010051376635457$i	$2.2838\times 10^{-8}$
	$I_2[f]$	$0.039933113640080 + 0.010051357751089$i
$10^4$	$4$	$-0.011029528472103 + 0.006076933181873$i	$ 3.2103\times 10^{-5}$
	$8$	$-0.011017821931600 + 0.006106493146584$i	$ 4.4922\times 10^{-7}$
	$16$	$-0.011017425972600 + 0.006106651469012$i	$2.2836\times 10^{-8}$
	$I_2[f]$	$-0.011017404229602 + 0.006106658450131$i

$\omega$	$N$	计算时间(FMM)	计算时间(Convention)	$\|Cu_N^{\omega}-u_N^{\omega}\|$
$10$	$4$	$ 2.4s$	$ 6.2s$	$ 0.0069$
	$8$	$ 3.9s$	$ 24.2s$	$ 0.0304$
	$16$	$ 14.1s$	$94.2s$	$ 0.054$
$100$	$4$	$ 10.8s$	$ 23.0s$	$ 2.91\times10^{-5}$
	$8$	$ 52.0s$	$ 93.6s$	$ 9.34\times10^{-5}$
	$16$	$ 179.6s$	$380.7.2s$	$ 1.84\times10^{-4}$
$160$	$4$	$ 14.7s$	$ 31.9s$	$ 1.18\times10^{-5}$
	$8$	$ 69.9s$	$ 115.8s$	$ 7.15\times10^{-5}$
	$16$	$ 405.0s$	$482.0s$	$ 5.38\times10^{-4}$
$200$	$4$	$ 15.9s$	$ 36.8s$	$ 8.68\times10^{-6}$
	$8$	$ 71.5s$	$ 125.1s$	$ 1.42\times10^{-5}$
	$16$	$ 427.0s$	$514.0s$	$ 3.00\times10^{-5}$