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计算数学  2018, Vol. 40 Issue (2): 171-190    DOI:
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Helmholtz方程有限差分方法概述
王坤1, 张扬1, 郭瑞2
1. 重庆大学数学与统计学院, 重庆 401331;
2. 石河子大学理学院数学系, 石河子 832003
FINITE DIFFERENCE METHODS FOR THE HELMHOLTZ EQUATION: A BRIEF REVIEW
Wang Kun1, Zhang Yang1, Guo Rui2
1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;
2. Department of Mathematics, Faculty of Sciences, Shihezi University, Shihezi 832003, China
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摘要 文章对最近二十年来Helmholtz方程有限差分方法方面的发展进行了概述.以相位误差为基础,文章分别对一维、二维、三维空间中该方面的研究结果进行了陈述,阐述了各种方法之间的差别与联系,特别展现了在高波数情况下不同差分格式对Helmholtz方程的计算效果,并且对高波数Helmholtz方程有限差分方法研究中现在存在的一些主要困难进行了讨论.
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关键词Helmholtz方程   高波数问题   有限差分方法   相位误差   数值色散     
Abstract: In this paper, a brief review on the development of the finite difference methods for the Helmholtz equation in the latest twenty years is presented. Based on the phase error, the results in 1D, 2D and 3D are given in this paper, the difference and relationship of different schemes are also shown, especially on the computational effect when applying to solve the Helmholtz equation with high wave numbers, moreover, some mainly problems existing in approximating the high wave number problems are discussed.
Key wordsHelmholtz equation   high wave number problem   finite difference method   phase error   numerical dispersion   
收稿日期: 2017-08-26;
基金资助:

中央高校基本科研业务费(资助号:106112017CDJXY100006)和重庆市基础科学与前沿技术研究专项(项目立项编号:cstc2017jcyjAX0231)资助;石河子大学自主资助支持校级项目ZZZC201611(2017-2018)和石河子大学高层次人才科研启动项目RCSX201733(2017-2020)资助.

引用本文:   
. Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190.
. FINITE DIFFERENCE METHODS FOR THE HELMHOLTZ EQUATION: A BRIEF REVIEW[J]. Mathematica Numerica Sinica, 2018, 40(2): 171-190.
 
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