波动谱元模拟中透射边界稳定性分析

无限域波动数值模拟中，人工边界的稳定性是获得可靠模拟结果的前提。具有高阶精度的谱元法和透射边界两者结合的数值模拟方案显示出较好的模拟精度和数值稳定性，然而，仍然存在数值失稳现象，其失稳机理和稳定条件尚不明确，相应的理论分析极为欠缺。该文针对透射边界在高阶谱元法中的稳定性，依据高阶谱单元中非等间距节点的周期延拓特点，通过构建内域和边界数值格式的向量形式来分析人工边界反射系数。进而保证边界对谱元法中存在的真实模态和虚假模态的反射系数均小于等于1，从而得到透射边界的稳定条件，其表现为无量纲边界参数和谱元参数之间的关系，其含义为透射边界人工波速与介质物理波速的比值限定在一定范围内。同时揭示了透射边界引发高频失稳的机理，即边界对谱元法中虚假模态的反复反射放大所致。最后采用数值实验验证了透射边界稳定条件。

STABILITY ANALYSIS OF TRANSMITTING BOUNDARY IN WAVE SPECTRAL ELEMENT SIMULATION

The stability of artificial boundary condition is the premise to obtain reliable simulation results in infinite wave simulation. The numerical schemes combining spectral element method and transmitting boundary with high order accuracy show better numerical accuracy and stability, but there still exits instability problem. The mechanism of instability and stability condition of transmitting boundary is still not known, and the corresponding theoretical analysis is extremely lacking. Aiming at the stability of transmitting boundary in spectral element method, according to the periodic extension property of the spectral element nodes that are not evenly spaced, the vector forms of spectral element method and transmitting boundary numerical scheme are presented. Based on the vector form, the reflection coefficient of transmitting boundary is deduced. Then the stability condition of transmitting boundary is obtained by guaranteeing the reflection coefficients of the real mode and all spurious modes in spectral element method are less than or equal to one. The stability condition shows the relationship between dimensionless parameter of transmitting boundary and that of spectral element method, that means the ratio of the artificial wave velocity to the physical wave velocity is limited to a certain range. Meanwhile, it is revealed that the mechanism of instability is the multi-reflection amplification of the spurious modes caused by the artificial boundary in the finite calculation area. Finally, the stability condition of transmitting boundary is verified by numerical experiments.