袁雨欣^1,2,3,4,,
李阿满^1,2,3,4,
胡婷^1,2,3,4,
郭鹏⁵,
刘洪^1,2,3,4,,
1. 中国科学院地质与地球物理研究所, 北京 100029
2. 中国科学院地球科学研究院, 北京 100029
3. 中国科学院油气资源研究院重点实验室, 北京 100029
4. 中国科学院大学, 北京 100049
5. 天津大学水利工程仿真与安全国家重点实验室, 天津 300072

基金项目: 国家自然科学基金项目（41630319），国家重点研发计划深地专项项目（2016YFC0601101）资助


详细信息

作者简介: 袁雨欣, 男, 1994年生, 在读博士研究生, 主要从事地震波数值模拟研究.E-mail:yuanyuxin@mail.iggcas.ac.cn

通讯作者: 刘洪, 男, 1959年生, 研究员, 主要从事地震波成像和油储地球物理研究.E-mail:liuhong@mail.iggcas.ac.cn

中图分类号: P631

收稿日期:2018-04-28
修回日期:2019-06-11
上线日期:2019-07-05

A multigrid-based preconditioner for solving the Helmholtz equation with average-derivative optimal scheme

YUAN YuXin^1,2,3,4,,
LI AMan^1,2,3,4,
HU Ting^1,2,3,4,
GUO Peng⁵,
LIU Hong^1,2,3,4,,
1. Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
2. Institutes of Earth Science, Chinese Academy of Sciences, Beijing 100029, China
3. Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
4. University of Chinese Academy of Sciences, Beijing 100049, China
5. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China


More Information

Corresponding author: LIU Hong,E-mail:liuhong@mail.iggcas.ac.cn

MSC: P631

--> Received Date: 28 April 2018
Revised Date: 11 June 2019
Available Online: 05 July 2019


摘要
摘要:有限差分法求解Helmholtz方程，依赖于两点：1差分格式的构造；2高效的求解算法.本文采用平均导数法离散Helmholtz方程.该差分格式有三点好处：1能适用于横纵不等间距采样；2在完全匹配层区域（PML），差分方程与微分方程逐点相容；3能将一个波长内的采样点数减少至少于4.求解离散的Helmholtz方程的算法一般分为直接法和迭代算法.直接法由于内存需求太大而无法适用于大规模问题；基于Krylov子空间的迭代方法结合多重网格预条件算法是一种快速高效求解方法，然而对于横纵不等间距采样（在多重网格中称为各向异性问题），经典的多重网格方法失效.本文分析了经典多重网格的三个重要组成部分：完全加权限制算子，点松弛技术以及双线性延拓算子，进而采用了半粗化技术代替全粗化技术，线松弛技术代替点松弛技术以及依赖差分算子的延拓算子代替双线性延拓算子，使得各向异性问题变得收敛；而且对于非均匀介质中-低频率的迭代问题，我们获得了较为满意的收敛速度.
关键词: 多重网格/
各向异性问题/
Helmholtz方程/
平均导数法

Abstract:An efficient finite-difference method for solving Helmholtz equation depends on two points:one is discrete scheme, the other is efficient algorithm. In this paper, we adopt the average-derivative scheme, which owns three advantages:Firstly, it can be applied to unequal directional sampling intervals for Helmholtz equation. Secondly, the scheme is pointwise consistent with Helmholtz equation in a perfect matched layer. And thirdly, it requires less than 4 grid points sampling per wavelength. To solve the discrete Helmholtz equation, which is extremely large and indefinite, direct methods cannot resolve well, and the Krylov subspace iterative methods, such as Bi-CGSTAB and GMRES combining a multigrid-based preconditioner, are good choices. However, the standard multigrid algorithm fails to converge when it encounters unequal directional sampling intervals, which is called anisotropy in multigrid. We analyze the most important three parts of standard multigrid:full weighting restriction operator, point relaxation methods and bilinear interpolation operator, and then we replace them with semi-coarsening, line relaxation and operator-dependent interpolation to make it convergent in anisotropic problems. Consequently, we obtain a satisfactory convergence speed for low and moderate frequency iterative problems in heterogeneous media.
Key words:Multigrid/
Anisotropic problem/
Helmholtz equation/
Average-derivative optimal scheme



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