Coupled Explicit Method and Multi-Transmitting Formula Analyzed for Wave Propagation

Researchers have coupled the explicit method (NEM), originally developed from the explicit lumped-mass finite-element method by Liu and Liao in 2009, with the multi-transmitting formula (MTF) to establish an integrated framework for one-dimensional wave propagation analysis. Within this framework, second- and fourth-order NEM-MTF coupled schemes were formulated using homogeneous spatial interpolation and exact time integration.

Through frequency-domain reflection analysis and numerical experiments, the investigation evaluated the mechanism of high-frequency oscillatory instability and identified corresponding stable parameter regions. The study further analyzed and interpreted the origin of unexpected local oscillatory instability. Additionally, long-term drift associated with low-frequency instability was examined numerically, with its stabilization remaining an area for future investigation.

To guide parameter selection, the effects of the NEM approximation order, MTF transmission order, and wave-speed ratio on computational accuracy were quantified. The proposed scheme was applied to simulate seismic wave propagation in a multilayered soil site, demonstrating computational performance, accuracy, and adaptability to layered media, while extending the scheme to multidimensional wave propagation problems requires additional investigation.

Subject of Research: One-dimensional wave propagation modeling using coupled explicit numerical methods and multi-transmitting formulas

Article Title: Development and stability analysis of an explicit method coupled with the multi-transmitting formula for one-dimensional wave propagation

Article References: Liu, H., Wang, Z., Yan, X., Xu, M., Wang, J., Pan, Q., & Jiang, J. (2026). Development and stability analysis of an explicit method coupled with the multi-transmitting formula for one-dimensional wave propagation. Earthquake Engineering and Engineering Vibration, 25(4), 925-949. https://doi.org/10.1007/s11803-026-2413-1

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DOI: 10.1007/s11803-026-2413-1

Keywords: wave propagation, explicit method, multi-transmitting formula, stability analysis, numerical modeling, finite element method, seismic wave propagation, layered media, Development, stability, analysis, explicit