## Efficient Numerical Methods for Integral Equations with Oscillatory Hankel Kernels

Wu Qinghua,1,2

 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

 Fund supported: the NSFC .  11701170the NSF of Hunan Province.  2017JJ3029the Young Core Teacher Foundation of Hunan Province

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.

Keywords： Oscillatory hankel kernels ; Highly oscillatory integral equations ; Fast multipole method

Wu Qinghua. Efficient Numerical Methods for Integral Equations with Oscillatory Hankel Kernels. Acta Mathematica Scientia[J], 2019, 39(3): 611-619 doi:

## 1 引言

$$$\label{eq:a1}\Delta u(x) + \omega^2u(x) = 0, \quad x \in \mathbb{R}^2\setminus \bar{\Omega},$$$

$$$\label{BIE1}u(x)=\int_{\Gamma}u(y)\frac{\partialG(x, y)}{\partialn(y)}-G(x, y)\frac{\partialu}{\partialn}(y){\rm d}s(y), \quad x\in \mathbb{R}^2\setminus \bar{\Omega},$$$

$$$\label{H01}G(x, y)=\frac{\rm i}{4}H_0^{(1)}(\omega|x-y|),$$$

$H_0^{(1)}$表示第一类Hankel函数, $n(y)$表示$y$点处的外法向量, $\Gamma$表示障碍物$\Omega$的边界.

$\begin{eqnarray}\label{Asy1}\Psi :=\left\{\begin{array}{ll} \frac{2}{\omega}\frac{\partial u^i}{\partial n}, \; & \mbox{in the illuminated region }(n\cdot d<0), \\ 0, & \mbox{in the shadow region }(n\cdot d>0).\end{array}\right. \end{eqnarray}$

$\begin{eqnarray}\label{bijjie} \frac{1}{\omega}\frac{\partial u}{\partial n}(x)=\Psi(x) + \omega \sum\limits_{m=1}^{n} [{\rm e}^{{\rm i}\omega x\cdot d_m}v_m^+(x, \omega)+{\rm e}^{-{\rm i}\omega x\cdot d_m}v_m^-(x, \omega)], x\in\Gamma, \end{eqnarray}$

$$$\label{BIE2}{\displaystyle\frac{1}{2}q(x)+\int_{\Gamma}\left(\frac{\partialG(x, y)}{\partialn(x)}+{\rm i}\eta G(x, y)\right) q(y){\rm d}s(y)=f(x), \quad x\in \Gamma, }$$$

## 2 快速多极边界元方法求解方程(1.6)

$\begin{eqnarray}\label{BIE2j} \nonumber &&\sum\limits_{m}\int_{\Gamma_m} \left(2\frac{\partial G(x, y)}{\partial n(x)}-2{\rm i}\omega G(x, y)\right)(\exp({\rm i}\omega x\cdot d_m)v_{m}^+ +\exp(-{\rm i}\omega x\cdot d_m)v^{-}_m){\rm d}s(y)\\ &&+\sum\limits_{m}\exp({\rm i}\omega x\cdot d_m)v_m^++\exp(-{\rm i}\omega x\cdot d_m)v_m^- \nonumber\\ &=&2\frac{\partial u^i}{\partial n}\frac{1}{\omega}-2{\rm i}u^i-\int_{\Gamma}\left(2\frac{G(x, y)}{n(x)}-2{\rm i}\omega G(x, y)\right)\Psi {\rm d}s(y)-\Psi.\end{eqnarray}$

$$$\label{J1} J_m( x)=\frac{1}{2\pi {\rm i}^m}\int_0^{2\pi}{\rm e}^{{\rm i}(x\cos(t)-mt)}{\rm d}t,$$$

$$$G(x, y)\approx \frac{\rm i}{4L}\textbf{h}_{\omega}(x, x_l)T_{\omega}(x_l, x_c)\textbf{g}_{\omega}(x_c, y),$$$

$$$h_{\omega}^k(x, x_l)={\rm e}^{{\rm i}\omega (x-x_l)\cdot S(\beta_k)}\label{h},$$$

$$$g_{\omega}^k(x_c, y)={\rm e}^{-{\rm i}\omega (x_c-y)\cdot S(\beta_k)}\label{g},$$$

$$$t_{\omega}^k(x_l, x_c)=\sum\limits_{j=-M}^M {\rm i}^{-j}H_{j}(\omega\|x_c-x_l\|){\rm e}^{{\rm i}j(\theta_{x_c-x_l}-\beta_k)}\label{T},$$$

$\begin{eqnarray}\nonumber \int_{\Gamma_j} G(x_k, y)q(y){\rm d}s(y)&\approx &\int_{\Gamma_j}\frac{\rm i}{4L}\textbf{h}_{\omega}(x_k, x_l)T_{\omega}(x_l, x_c)\textbf{g}_{\omega}(x_c, y)q(y){\rm d}s(y)\\&=&\frac{\rm i}{4L}\textbf{h}_{\omega}(x_k, x_l)T_{\omega}(x_l, x_c)M_{\omega}(x_c), \label{M1}\end{eqnarray}$

$$$\label{M2M}M_{\omega}^k (x_{c'})={\rm e}^{-{\rm i}\omega (x_{c'}-x_c)}M_{\omega}^k (x_{c}).$$$

$$$L_{\omega}(x_l)=T_{\omega}(x_l, x_c)M_{\omega}(x_c),$$$

$$$\label{L2L}L_{\omega} (x_{l'})={\rm e}^{{\rm i}\omega (x_{l}-x_{l'})}L_{\omega} (x_{l}),$$$

## 3 Clenshaw-Curtis Filon方法计算高振荡积分

$$$\label{Int2} I[f]=\int_a^b f(x){\rm e}^{{\rm i}\omega g(x)} {\rm d}x.$$$

$\omega\gg 1$时,被积函数是高振荡的,经典的数值积分法计算这类积分是很困难的,参看文献[12-13, 22].

$\begin{eqnarray}I_N[f]:=\int_{-1}^1 Q_N(x){\rm e}^{{\rm i}\omega x}{\rm d}x=\sum\limits_{n=0}^{N}{''}a_{n, N}(f)W_n, \end{eqnarray}$

$$$\label{Chebyrec} 2W_n=\rho_{n+1}-\rho_{n-1}, \quad n\geq 2,$$$

$$$\label{ditui} W_0=\sigma_0 , \quad W_n=\sigma_n-\frac{n}{{\rm i}\omega}\rho_{n}, \quad n\geq 1,$$$

$$$\label{gongshi} 2\sigma_n-\frac{2n}{{\rm i}\omega}\rho_n=\rho_{n+1}-\rho_{n-1}, \quad n\geq 2.$$$

$\bullet$如果$g{'}(x)\neq 0$, $x\in[a, b]$,利用变量替换$\tau=g(x) , $$g^{-1}(\tau)\in C^{\infty},且 \begin{eqnarray} I^{[a, b]}[f]=I^{[g(a), g(b)]}[F], \; \mbox{其中}\ F=(f\circ g^{-1})|(g^{-1})'|, \end{eqnarray} 则(3.1)式可以用CCF方法计算. \bullet如果存在\xi\in [a, b]使得g'(\xi)=g''(\xi)= \cdots = g^{(r)}(\xi)=0, g^{(r+1)}(\xi) \neq 0$$g{'}(x)\neq 0$, $x\neq \xi$.不失一般性,假设$g^{(r+1)}(\xi)>0$.在每个区间上利用变量替换可得

 $\omega$ $m$ $I_m[f]$ $|I1[f]-I_m[f]|$ $10^3$ $4$ $-0.012459303385353 + 0.020469089205557$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

## 4 数值算例

$Cu_N^{\omega}$经典配置方法得到的解, $u_N^{\omega}$为本文方法计算得到的解,计算结果见表 3.

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

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