波动数值模拟中透射边界的稳定实现

STABLE IMPLEMENTATION OF TRANSMITTING BOUNDARY IN NUMERICAL SIMULATION OF WAVE MOTION

从波动能量在计算区内累积增大的观点出发 ,通过简单的一维弹性波模型 ,系统地阐明了在近场波动数值模拟中透射边界两类数值失稳———“高频振荡”和“零频飘移”的机理 :前者源于对波动数值模拟无意义的高频波动在人工边界上的放大和波在有限计算区内多次反射产生的反复放大 ;后者则源于透射边界允许零频和接近零频的分量不断进入计算区 .由此提出了稳定实现透射边界的完整方案包括两项简单措施 :第一 ,在全部计算区内按文中建议的方法注入小阻尼 ,以消除高频振荡 ;第二 ,给出一种具有明确物理意义的消除零频飘移的算子算法 .最后 ,提供了三维波源问题和散射问题的详细数值试验结果 .

The numerical instability resulted from the transmitting boundary in numerical simulation of the near field wave motion is systematically discussed from a unified viewpoint that wave energy within the computational region might accumulate and increase unceasingly. Mechanisms of two types of the instability, namely, the high frequency oscillation and the zero frequency drift are then clarified as follows: the former results from amplification of high frequency waves, which are meaningless for numerical simulation of wave motion by finite elements or finite differences, reflected from the transmitting boundary, and from repeated amplifications of multiple reflection of the waves within the finite computational region; the latter results from that the boundary allows the wave components of zero or approximately zero frequencies entering the computational region continuously. Therefroe, a practical and complete scheme is suggested for stable implementation of the transmitting boundary that includes two simple measures: (1) injecting small amount of damping into the entire computational region by the procedure presented in the paper to eliminate the high frequency oscillation, and (2) using an operator algorithm of clear physical meaning to eliminate the zero frequency drift. The implementation scheme is finally tested in detail by numerical experiments of three dimensional wave motions including a source problem and a scattering problem.