Abstract Application of local artificial boundary conditions in high-efficient spectral element method (SEM) is a problem that needs further study, where difficulties focus on how to design an appropriate numerical scheme of artificial boundaries that is in good accordance with the unequally-spaced grid nodes and high-order property of the element. Liao's local artificial boundary, i.e. multi-transmitting formula (MTF), provides a convenient way of implementing arbitrary-order displacement-type boundaries because it is defined directly in discrete form and can be transformed into numerical schemes by using merely some interpolation approaches. A new set of numerical schemes of MTF is developed and its accuracy and stability are thoroughly discussed in the application of spectral-element simulation of 1D and 2D seismic wave propagation. The new MTF scheme cannot be written into a binomial form like the traditional one does. This fact has a significant influence on numerical properties of high-order boundary, but merely slightly affects the performance of lower-order boundaries which are daily used in practical simulations. Numerical examples compared with viscous-spring or perfectly matched layer (PML) boundary exhibits that stable and quite good accuracy can be achieved by using the proposed method.

中文翻译：

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一维和二维地震波建模中多传输公式的谱元公式及其准确性和稳定性

摘要 局部人工边界条件在高效谱元法（SEM）中的应用是一个需要进一步研究的问题，其难点集中在如何设计一个合适的、与不等距网格相吻合的人工边界数值方案。节点和元素的高阶属性。Liao 的局部人工边界，即多传递公式（MTF），提供了一种实现任意阶位移型边界的便捷方法，因为它直接以离散形式定义，并且可以仅使用一些插值方法将其转换为数值方案。开发了一套新的MTF数值方案，并深入讨论了其在一维和二维地震波传播谱元模拟中的应用的准确性和稳定性。新的 MTF 方案不能像传统的那样写成二项式形式。这一事实对高阶边界的数值性质有显着影响，但对实际模拟中日常使用的低阶边界的性能仅略有影响。与粘性弹簧或完美匹配层 (PML) 边界相比的数值示例表明，使用所提出的方法可以实现稳定且相当好的精度。