Abstract 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 preconditoner 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.

中文翻译：

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具有预处理迭代求解器的 3D Helmholtz 方程的新有限差分格式

摘要 在本文中，我们针对 3D Helmholtz 问题提出了一种新的有限差分格式，该格式紧凑且精度为四阶。与标准的紧凑四阶方案不同，新方案是在最小化数值色散的基础上专门建立的，通过用27个点的加权平均值逼近方程的零阶项。为确定最优权重参数，提出一个优化问题，然后采用基于色散方程的奇异值分解方法进行处理。对于所提出的方案，通过巧妙地将 3D 误差方程分解为几个 1D 差分问题，努力推导出解的唯一性和收敛性。为了解决由差分离散化产生的稀疏和大尺寸的线性系统，我们开发了基于移位拉普拉斯和 3D 全粗化多重网格预处理的 Bi-CGSTAB 迭代求解器。移位拉普拉斯算子用于通过所提出的紧凑四阶方案生成具有离散化的预条件子，而构建具有基于矩阵的扩展算子的全粗化多重网格来近似预条件子的逆。最后，给出了数值例子来证明新的差分方案和预处理求解器的效率。而具有基于矩阵的扩展算子的全粗化多重网格被构建为近似于预处理器的逆。最后，给出了数值例子来证明新的差分方案和预处理求解器的效率。而具有基于矩阵的扩展算子的全粗化多重网格被构建为近似于预处理器的逆。最后，给出了数值例子来证明新的差分方案和预处理求解器的效率。