PML and CFS-PML boundary conditions for a mesh-free finite difference solution of the elastic wave equation

Mesh-free finite difference (FD) methods can improve the geometric flexibility of modeling without the need for lattice mapping or complex meshing process. Radial-basis-function-generated FD is among the most commonly used mesh-free FD methods and can accurately simulate seismic wave propagation in the non-rectangular computational domain. In this paper, we propose a perfectly matched layer (PML) boundary condition for a mesh-free FD solution of the elastic wave equation, which can be applied to the boundaries of the non-rectangular velocity model. The performance of the PML is, however, severely reduced for near-grazing incident waves and low-frequency waves. We thus also propose the complex-frequency-shifted PML (CFS-PML) boundary condition for a mesh-free FD solution of the elastic wave equation. For two PML boundary conditions, we derive unsplit time-domain expressions by constructing auxiliary differential equations, both of which require less memory and are easy for programming. Numerical experiments demonstrate that these two PML boundary conditions effectively eliminate artificial boundary reflections in mesh-free FD simulations. When compared with the PML boundary condition, the CFS-PML boundary condition results in better absorption for near-grazing incident waves and evanescent waves.