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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (5): 726-742.doi: 10.3969/j.issn.1005-3085.2015.05.011

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求解Helmholtz方程的快速算法(英)

冷  伟   

  1. 中国科学院科学与工程计算国家重点实验室，北京 100190
  • 收稿日期:2015-01-01 接受日期:2015-08-25 出版日期:2015-10-15 发布日期:2015-12-15
  • 基金资助:
    国家863项目 (2012AA01A309).

A Fast Propagation Method for the Helmholtz Equation

LENG Wei   

  1. State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing 100190
  • Received:2015-01-01 Accepted:2015-08-25 Online:2015-10-15 Published:2015-12-15
  • Supported by:
    The National 863 Project of China (2012AA01A309).

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摘要： 本文提出了求解Helmholtz方程的一个新的快速算法．该算法是建立在有重叠区域的区域分解算法之上的．该算法首先对求解区域进行层次的区域分解，然后建立了各层次的子区域上的入射波到出射波的映射，最后通过层次的传播波的信息，得到Helmholtz方程的解．该方法具有计算复杂度小、适合大规模并行计算的优点，数值实验表明， 该方法能够有效的并行求解有上亿自由度的二维Helmholtz方程．

关键词: Helmholtz方程, 有限差分, 快速算法, 区域分解, 完全匹配层

Abstract:

A fast method is proposed for solving the high frequency Helmholtz equation. The building block of the new fast method is an overlapping domain decomposition method for layered medium. In the new fast method, the computation domain is firstly decomposed hierarchically into many subdomains on different levels. Then the mapping from incident waves to out-going waves on all the subdomains are set up. Finally, the wave propagates on the subdomain boundaries on different levels to reach the solution to the Helmholtz equation. The new fast method is of low complexity, and suitable for parallel computing. Numerical experiments show that with the new fast method, 2D Helmholtz equations with half billion unknowns could be solved efficiently on massively parallel machines.

Key words: Helmholtz equation, finite difference, fast method, domain decomposition method, PML

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