无网格广义有限差分格式在波动方程中的稳定性研究

IF 1.3 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Frontiers in Applied Mathematics and Statistics Pub Date : 2023-07-11 DOI:10.3389/fams.2023.1214022
G. Tinoco-Guerrero, F. Domínguez-Mota, J. A. Guzmán-Torres, Ricardo Román-Gutiérrez, J. Tinoco-Ruiz
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引用次数: 0

摘要

在设计和实现数值格式时,必须考虑所用方法的稳定性。先前的研究对应用于平流和扩散方程的广义有限差分方法的稳定性给出了不同的结果。近年来,研究人员探索了一种广义有限差分方法来求解非矩形和高度不规则区域上的平流-扩散方程,该方法使用凸的逻辑矩形网格。本文研究了应用于波动方程数值解的广义有限差分格式的稳定性,该方程在高度不规则域的点云上求解。这项工作中提出的稳定性分析为获得稳定和令人满意的结果所需的适当离散化提供了重要的见解。所提出的显式方案是条件稳定的,而隐式方案是无条件稳定的。值得注意的是,本文提出的稳定性分析适用于空间中至少为二阶的任何方案,而不仅仅是所提出的方法。所提出的方案为数值求解波动方程提供了有效的方法,特别是对于高度不规则的区域。通过验证该方案的稳定性,为该领域的进一步研究奠定了基础。
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Study of the stability of a meshless generalized finite difference scheme applied to the wave equation
When designing and implementing numerical schemes, it is imperative to consider the stability of the applied methods. Prior research has presented different results for the stability of generalized finite-difference methods applied to advection and diffusion equations. In recent years, research has explored a generalized finite-difference approach to the advection-diffusion equation solved on non-rectangular and highly irregular regions using convex, logically rectangular grids. This paper presents a study on the stability of generalized finite difference schemes applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The stability analysis presented in this work provides significant insights into the proper discretizations needed to obtain stable and satisfactory results. The proposed explicit scheme is conditionally stable, while the implicit scheme is unconditionally stable. Notably, the stability analyses presented in this paper apply to any scheme which is at least second order in space, not just the proposed approach. The proposed scheme offers effective means of numerically solving the wave equation, particularly for highly irregular domains. By demonstrating the stability of the scheme, this study provides a foundation for further research in this area.
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来源期刊
Frontiers in Applied Mathematics and Statistics
Frontiers in Applied Mathematics and Statistics Mathematics-Statistics and Probability
CiteScore
1.90
自引率
7.10%
发文量
117
审稿时长
14 weeks
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