中文摘要：

在哈密尔顿体系下，提出气体声波传播的一种新的谐振子模型，并引入群论确定气体声波传播过程中的分子振动模式、能级简并。新模型将气动声学声传播问题与分子振动关联起来。由于发展高效的薛定谔方程的数值计算方法，有利于联系分子的性质来解释声的传播。本文从此出发，用二阶有限差分格式和生成函数法构造的二阶辛格式分别计算一维定态谐振子势场和含时谐振子势场的薛定谔方程，分析了数值解的误差以及传播能量误差。结果表明辛算法具有明显的优势。

英文摘要：

A new harmonic oscillator model for solving aeroacoustics wave propagation in the Hamilton system is expounded, and the group theory is introduced to determine molecular vibrational modes and degeneracy of energy levels during acoustic wave propagation in the air. In the new model, the problem of aeroacoustics wave propagation is related to the molecular vibration. The development of efficient numerical method for the SchrSdinger equation is favorabe for explaining aeroacoustics wave propagation by contacting the molecular characteristic in the Hamilton system. Based on this theory, in this paper, we use second-order finite difference scheme and the second-order symplectic scheme constructed by the generating function method to calculate the one-dimensional stationary state harmonic oscillator and time-harmonic oscillator Schr5dinger equation, the error of the numerical solution and the error of dissemination energy are investigated. Results show that the symplectic algorithm has obvious advantages.