Trace transfer-based diagonal sweeping domain decomposition method for the Helmholtz equation: Algorithms and convergence analysis
Trace transfer-based diagonal sweeping domain decomposition method for the Helmholtz equation: Algorithms and convergence analysis
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DOI：
10.1016/j.jcp.2022.110980
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发表时间：
2020-12
期刊：
影响因子：
--
通讯作者：
W. Leng;L. Ju
中科院分区：
文献类型：
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作者：
W. Leng;L. Ju
利用完全匹配层(PML)和源转移技术，最近发展了对角扫描区域分解方法(DDM)来求解Rn中的高频Helmholtz方程，该方法使用2n个对角方向的扫描和棋盘区域分解。虽然这种对角线扫描DDM本质上是乘法的，但当与流水线处理相结合时，它非常适合于具有多个右端的Helmholtz问题的并行计算，因为每次扫描中的连续步数比子域的数目少得多。本文提出并分析了一种基于迹传递的对角扫描DDM算法。对于相邻子域之间的信息传递，从源转移改变为迹线转移的主要优点是，所得到的对角扫描变得更容易分析和实现，并且更有效，因为转移的迹线在相邻子域之间仅具有2n个基本方向，而转移的源来自总共3n个−1个基点方向和角点方向。我们严格证明了所提出的对角扫描DDM不仅给出了恒定媒质情况下整体PML问题的精确解，而且在双层媒质情况下最多只需一轮额外的对角扫描即可得到该问题的精确解，为该方法奠定了理论基础。通过大量的二维和三维实验，验证了DDM作为直接求解器或预条件器的性能和并行可伸缩性。
By utilizing the perfectly matched layer (PML) and source transfer techniques, the diagonal sweeping domain decomposition method (DDM) was recently developed for solving the high-frequency Helmholtz equation in R n, which uses 2 n sweeps along respective diagonal directions with checkerboard domain decomposition. Although this diagonal sweeping DDM is essentially multiplicative, it is highly suitable for parallel computing of the Helmholtz problem with multiple right-hand sides when combined with the pipeline processing since the number of sequential steps in each sweep is much smaller than the number of subdomains. In this paper, we propose and analyze a trace transfer-based diagonal sweeping DDM. A major advantage of changing from source transfer to trace transfer for information passing between neighbor subdomains is that the resulting diagonal sweeps become easier to analyze and implement and more efficient, since the transferred traces have only 2 n cardinal directions between neighbor subdomains while the transferred sources come from a total of 3 n− 1 cardinal and corner directions. We rigorously prove that the proposed diagonal sweeping DDM not only gives the exact solution of the global PML problem in the constant medium case but also does it with at most one extra round of diagonal sweeps in the two-layered media case, which lays down the theoretical foundation of the method. Performance and parallel scalability of the proposed DDM as direct solver or preconditioner are also numerically demonstrated through extensive experiments in two and three dimensions.