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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   

  1. 华南师范大学数学科学学院, 广州 510631
  • 收稿日期:2017-01-06 出版日期:2017-12-15 发布日期:2017-11-13
  • 通讯作者: 黎稳
  • 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.

骆其伦, 黎稳. 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.

Luo Qilun, Li Wen. 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.

THE PRECONDITIONER FOR LINEAR EQUATIONS DISCRETIZED FROM TWO-DIMENSIONAL HELMHOLTZ EQUATION BY COMBINED COMPACT DIFFERENCE SCHEMES

Luo Qilun, Li Wen   

  1. School of Mathematical Sciences South China Normal University, Guangzhou 510631, China
  • Received:2017-01-06 Online:2017-12-15 Published:2017-11-13
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.
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.

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