# 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

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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.

1. 引言

$\left\{\begin{array}{l}\frac{{\partial }^{2}u}{\partial {t}^{2}}+{a}^{2}\frac{{\partial }^{4}u}{\partial {x}^{4}}=0,00\\ {u|}_{t=0}=\phi \left(x\right),{\frac{\partial u}{\partial t}|}_{t=0}=\psi \left(x\right),0 (1.1)

2. 差分格式的构造

2.1. 三层差分显格式的构造

。 (2.1)

2.2. 三层差分隐格式的构造

。(2.2)

2.3. 人工边界

。 (2.3)

。 (2.4)

。 (2.5)

3. 差分格式的稳定性分析

3.1. 显格式的稳定性

。其特征方程的解为

3.2. 隐格式的稳定性

，则上述方程组可写成：

，将其带入上述方程组并消去公因子，得到

，上述方程组化为

4. 数值实验

Figure 1. Numerical solution obtained by the explicit scheme with different grid ratio, left:，right:

Table 1. Experiment convergence rate for the discrete maximum norm and -norm

5. 结论

NOTES

*通讯作者。

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