计算数学
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计算数学  2017, Vol. 39 Issue (4): 407-420    DOI:
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二维Helmholtz方程的联合紧致差分离散方程组的预处理方法
骆其伦, 黎稳
华南师范大学数学科学学院, 广州 510631
THE PRECONDITIONER FOR LINEAR EQUATIONS DISCRETIZED FROM TWO-DIMENSIONAL HELMHOLTZ EQUATION BY COMBINED COMPACT DIFFERENCE SCHEMES
Luo Qilun, Li Wen
School of Mathematical Sciences South China Normal University, Guangzhou 510631, China
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摘要 对于二维的Helmholtz方程,本文用联合紧致差分格式(CCD)离散,该差分格式具有六阶精度,三点差分和隐式的特点.本文基于CCD格式离散得到的线性系统和循环矩阵的快速傅里叶变换,提出了一种循环型预处理算子用于广义极小残量迭代算法(GMRES).给出了循环型预处理子的求解算法,证明了该预处理算子能使迭代算法具有较快的收敛速度.本文还与其他算法的预处理算子作比较,数值结果表明本文提出的循环型预处理算子具有更好的稳定性,并且对于较大的波数k,收敛速度也更快.
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关键词Helmholtz方程   联合紧致差分格式   广义极小残量法   循环型预处理算子     
Abstract: Combined compact difference (CCD) schemes are used to discretize two-dimension Helmholtz equations, the fundamental features of this scheme are given as follow:three point, implicit and sixth-order accuracy. In this paper, a circulant-like preconditioner is proposed for the generalized minimal residual method (GMRES) iterative algorithm based on the discretized CCD linear system and the fast fourier transform of the circulant matrix. We also give an algorithm for solving circulant-like preconditioner which is shown to have a faster convergence. Moreover, the numerical results show that the circulant-like preconditioner for GMRES is more stable and got the faster convergence than other preconditioner when the wave number k is large.
Key wordsHelmholtz equation   combined compact difference schemes   generalized minimal residual method   circulant-like preconditioner   
收稿日期: 2017-01-06;
基金资助:

该项目受国家自然基金(11671158,11771159),广东省普通高校省级重大项目(2016KZDM025)与创新团队建设项目(2015KCXTD007)资助.

通讯作者: 黎稳     E-mail: liwen@scnu.edu.cn
引用本文:   
. 二维Helmholtz方程的联合紧致差分离散方程组的预处理方法[J]. 计算数学, 2017, 39(4): 407-420.
. THE PRECONDITIONER FOR LINEAR EQUATIONS DISCRETIZED FROM TWO-DIMENSIONAL HELMHOLTZ EQUATION BY COMBINED COMPACT DIFFERENCE SCHEMES[J]. Mathematica Numerica Sinica, 2017, 39(4): 407-420.
 
[1] Fix G J, Marin S P. Variational methods for underwater acoustic problems[J]. Journal of Computational Physics, 1978, 28(2):253-270.
[2] Chen I L, Chen J T, Liang M T. Analytical study and numerical experiments for radiation and scattering problems using the CHIEF method[J]. 2001, 248(5):809-828.
[3] Ciavaldini J F, Tournemine G. A finite element method to compute stationary steady flows in the hodograph plane[J]. Journal of the Indian Mathematical Society, 1977, 41(1-2):69-82.
[4] Plessix, Mulder. Frequency-domain finite-difference amplitude-preserving migration[J]. Geophysical Journal International, 2004, 157(3):975-987.
[5] Erlangga Y A. Advances in Iterative Methods and Preconditioners for the Helmholtz Equation[J]. Archives of Computational Methods in Engineering, 2008, 15(1):37-66.
[6] J Berkhout A, Pao Y. Seismic Migration Imaging of Acoustic Energy by Wave Field Extrapolation[J]. Journal of Applied Mechanics, 1982, 49(3):682-683.
[7] Cai X C, Casarin M A, Elliott F W, et al. Overlapping Schwarz Algorithms For Solving Helmholtz's Equation[J]. Contemporary Mathematics, 1998, 218:391-399.
[8] 龙毅, 徐军, 朱汉清. 规则区域上Helmholtz方程的一种快速算法[J]. 电子科技大学学报(微波与激光技术专辑), 1999, 28(4):383-387.
[9] 龚东山, 丁方允. 关于大波数k的Helmholtz方程有限元解的误差估计与差量分析[J]. 兰州大学学报, 2003, 38(5):13-23.
[10] 丁方允, 吕涛涛. 二维Helmholtz方程非线性边值问题的边界元分析[J]. 兰州大学学报, 1994, 30(2):25-30.
[11] 丁方允. 三维Helmholtz方程Dirichlet问题的边界元法及其收敛性分析[J]. 兰州大学学报, 1995, 31(3):30-38.
[12] Chu P C, Fan C. A Three-Point Combined Compact Difference Scheme[J]. Journal of Computational Physics, 1998, 140(2):370-399.
[13] Liu Jun. Efficient Preconditioners for the Helmholtz Equation Discretized by Combined Compact Difference Method[D]. 华南师范大学硕士学位论文, 2011.
[14] 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. 浏览
[15] Saad Y. Iterative Methods for Sparse Linear Systems[M]. Society for Industrial & Applied Mathematics. 2003.
[16] James W. Cooley, John W. Tukey. An Algorithm for the Machine Calculation of Complex Fourier Series[J]. Mathematics of Computation, 1965, 19(90):297-301.
[17] Davis P J. Circulant matrices[M]. American Mathematical Soc., 2012.
[18] XQ Jin, YM. Wei, Numerical Linear Algebra and Its Applications[M]. Science Press, Beijing, 2004.
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