High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
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DOI：
10.1515/cmam-2022-0002
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发表时间：
2022-11
影响因子：
1.3
通讯作者：
Z. Wang;Y. Ge;Hai-wei Sun
中科院分区：
文献类型：
--
作者：
Z. Wang;Y. Ge;Hai-wei Sun
摘要本文提出了求解二维和三维Helmholtz方程的六阶紧致差分格式。首先，用二阶导数的六阶紧差分算子逼近拉普拉斯算子。同时，在原微分方程的基础上，提出了六阶紧致差分格式。然而，该方案的截断误差明显依赖于未知量，源函数和波数。因此，我们对六阶紧致格式的截断误差进行了修正，得到了精度更高的改进六阶紧致格式。从理论上证明了改进方法的收敛性和稳定性。最后通过数值试验验证了改进格式的精度。
Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.