A unified approach to high-order compact finite difference schemes for 3D Poisson equations

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics Pub Date : 2025-09-09 DOI:10.1016/j.cam.2025.117045

Kang Fu , Hongling Hu , Zhilin Li , Kejia Pan

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引用次数: 0

Abstract

A unified approach to high-order compact finite difference schemes for solving three-dimensional Poisson equations is derived. Notice that all high order compact finite difference schemes are linear combinations of the solution values at grid points and source terms at selected points. In the unified strategy, a parameter

γ

is introduced for the linear combination of the solution values at the compact finite difference stencil. In our approach, carefully chosen

γ

’s lead to different fourth-order schemes. Carefully chosen linear combination of the source term at grid points will lead to different fourth-order schemes. If additional points such as the middle points are used, sixth-order schemes can be achieved if the solutions with certain symmetries. The convergence proof is provided for the new unified scheme based on the

M

-matrix condition along with non-trivial numerical examples.

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三维泊松方程高阶紧致有限差分格式的统一方法

导出了求解三维泊松方程的高阶紧致有限差分格式的统一方法。注意，所有高阶紧致有限差分格式都是网格点的解值和选定点的源项的线性组合。在统一策略中，对紧致有限差分模板处解值的线性组合引入了参数γ。在我们的方法中，精心选择的γ会导致不同的四阶格式。在网格点处仔细选择源项的线性组合将导致不同的四阶格式。如果使用中间点等附加点，如果解具有一定的对称性，则可以实现六阶格式。在m -矩阵条件下给出了新的统一格式的收敛性证明，并给出了非平凡的数值算例。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational and Applied Mathematics 数学-应用数学

CiteScore

5.40

自引率

4.20%

发文量

437

审稿时长

3.0 months

期刊介绍： The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest. The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.