Stable Implementation of Transmitting Boundary in Numerical Simulation of Wave Motion

Abstract

Abstract 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. Therefrom, 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.