Spectral-element formulation of multi-transmitting formula and its accuracy and stability in 1D and 2D seismic wave modeling

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.