高振荡Hankel核积分方程的高效数值算法

Abstract

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

CLC Number:

O241.83

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

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References 25

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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}$

Efficient Numerical Methods for Integral Equations with Oscillatory Hankel Kernels

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