A high-order perfectly matched layer scheme for second-order spectral-element time-domain elastic wave modelling

Abstract

In this study, we develop a new high-order perfectly matched layer (PML) formulation for the 3D spectral-element time-domain (SETD) method to solve the second-order elastic wave equations in terms of displacements. The high-order PML consists of multiple well-known frequency-dependent complex stretching functions, leading to the complicated temporal convolution which requires large computational resources. To avoid the evaluation of convolution, we deal with the high-order stretching functions by introducing the recursive auxiliary first-order differential equations which can be easily solved in time domain. In addition, an equivalent concept of PML is adopted for absorbing waves, preserving the original form of wave equations in physical domain. Finally, we obtain a high-order, non-convolutional and unchanged-form PML formulation, which can be easily incorporated into the SETD codes designed for the physical domain, to make them applicable for the second-order wave equations in the unbounded media. Numerical experiments demonstrate the accuracy and stability of the high-order PML through the comparison with the reference solution. It is found that the high-order PML provides a much better absorption performance than the first-order PML, particularly for the grazing-incidence waves and surface waves in the large-scale domain. We also give a fluid-filled borehole model to illustrate that the high-order PML works well for absorbing the complex guided waves in the presence of fluid-solid interaction.