A new finite difference scheme for the 3D Helmholtz equation with a preconditioned iterative solver
A new finite difference scheme for the 3D Helmholtz equation with a preconditioned iterative solver
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具有预条件迭代求解器的 3D 亥姆霍兹方程的新有限差分格式
DOI：
10.1016/j.apnum.2020.11.023
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发表时间：
2021-03
影响因子：
2.8
通讯作者：
Dongsheng Cheng
中科院分区：
文献类型：
--
作者：
Tingting Wu;Yuran Sun;Dongsheng Cheng
本文对三维Helmholtz问题提出了一种新的有限差分格式，该格式是紧致的，具有四阶精度。与标准的紧致四阶格式不同，新格式是基于最小化数值频散的原则，通过对方程的零阶项进行27点的加权平均来逼近。为了确定最优权值参数，首先建立了一个优化问题，然后利用基于色散方程的奇异值分解方法进行求解。该格式巧妙地将三维误差方程分解为若干一维差分问题，从而得到了解的唯一性和收敛性。为了解决所产生的线性系统源于差分离散，这是稀疏和大尺寸的，我们开发了一个双CGSTAB迭代求解器的基础上的预处理的移位拉普拉斯和三维全粗多重网格。移位拉普拉斯算子用于生成预条件子，并通过所提出的紧凑四阶格式进行离散化，而基于矩阵的延拓算子的全粗化多重网格用于近似预条件子的逆。最后，数值算例验证了新差分格式和预处理方法的有效性。
In this paper, we propose a new finite difference scheme for the 3D Helmholtz problem, which is compact and fourth-order in accuracy. Different from a standard compact fourth-order one, the new scheme is specially established based on minimizing the numerical dispersion, by approximating the zeroth-order term of the equation with a weighted-average for the values at 27 points. To determine optimal weight parameters, an optimization problem is formulated and then dealt with the singular value decomposition method based on the dispersion equation. For the proposed scheme, by skillfully splitting the 3D error equation into several 1D difference problems, the solution's uniqueness and convergence are derived with an effort. To solve the resulting linear system stemming from difference discretization, which is sparse and large-sized, we develop a Bi-CGSTAB iterative solver based on the preconditioning of shifted-laplacian and 3D full-coarsening multigrid. The shifted-laplacian is used to generate the preconditioner with a discretization by the proposed compact fourth-order scheme, while the full-coarsening multigrid with matrix-based prolongation operators is built to approximate the inverse of the preconditioner. Finally, numerical examples are presented to demonstrate the efficiency of the new difference scheme and the preconditioned solver.
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