简支梁横振动方程的稳定差分格式的构造

Construction of Stable Difference Schemes for Transverse Vibration Equations of Simply Supported Beam

摘要: 这篇文章考虑简支梁的横振动方程的有限差分格式的构造：基于Taylor展开获得求解四阶问题的三层差分显格式和隐格式，它们的局部截断误差均为Ο（τ²+h²）。为了离散简支梁两端扰度为零这一边界条件而引入了人工边界条件，根据离散Fourier分析证明了当网比r=a²τ²/h⁴≤1/8时显格式稳定，而隐格式是绝对稳定的。数值实验证明了理论结果。

Abstract: In this paper, we consider the construction of stable difference schemes for the transverse vibration equations for simply supported beam: Based on Taylor expansion, three-level difference explicit and implicit schemes are developed for fourth order problem, and the order of local truncation error for these two schemes is proven to be Ο（τ²+h²）. An artificial boundary condition is introduced in order to discrete the boundary condition that the perturbation vanishes at two ends of the simply supported beam. Owing to the discrete Fourier analysis, the explicit scheme is proven to be stable when the grid ratio r=a²τ²/h⁴≤1/8, and the implicit scheme is proven to be absolutely stable. Numerical experiments confirm the theoretical results.