Abstract: Multi-transmitting formula (MTF, referred to as transmitting boundary for simplicity) is a kind of widely-used artificial boundary condition in the numerical simulation of near-field wave motion. It has advantages as very simple definition and formulations, adjustable accuracy and excellent versatility. However, higher-order MTFs suffer from drift problem from time to time when they are applied in finite-element simulations. There have been several ways of suppressing the drift instability of MTF, but those ways are usually accompanied by remarkable loss of accuracy. This work reports a new modified MTF scheme with a drift-elimination factor, which can effectively control the drift problem of MTF at a high level of accuracy. This approach keeps the first-order transmitting term of MTF unchanged and only modifies those terms regarding higher-order transmitting errors, thus the loss of accuracy caused by the modification is greatly reduced. Meanwhile, the added drift-elimination factor ensures the satisfaction of GKS stability criterion in the case of zero-frequency and zero-wavenumber wave energies, which gives a theoretical support for the control of drift instability. A higher-order unified expression of the proposed approach is further summarized, in which the traditional Zhou-Liao’s modified MTF with drift-elimination factor can be seen as a special case of this work. A comparison analysis of the reflection coefficient of different boundary methods shows that the proposed approach not only has superiority in accuracy, but also adapts to a much wider range of the value of drift-elimination factor. Finally, two numerical tests in the context of finite-element simulation of SH wave propagation validate the effectiveness of the proposed approach in both controlling drift problem and maintaining the accuracy of higher-order MTFs. The drift-elimination factor in the proposed approach has little influence on the absorption capacity of those waves impinging the artificial boundary under normal incidence or over small incident angles, in which most of the wave energies have been taken into account.