A new generalized k-space (GkS) method for transient elastodynamic scattering problems

Abstract

A conventional approach to simulating transient elastic wave propagation in inhomogeneous media has been the finite-difference (FD) method. However, the FD method requires a large number of grids in order to obtain accurate results. This is because in conventional FD schemes, second-order (sometimes higher-order) differences are used to approximate the spatial derivatives. In this work, a new generalized k-space (GkS) method is described for elastodynamic scattering problems. From its integral representation in spatial-frequency (r-f ) domain, a local equation is derived for the displacement field in spectral-frequency (k-f ) domain. This equation becomes a time-convolution equation in spectral-time (k-t) domain. Using two temporal propagators, compressional and shear, this time-convolution equation can be converted into two time-stepping equations, which become much easier to solve. Hence, at each time step, the solution is first obtained in the k-t domain, and then transformed to the r-t domain by using spatial FFT. Since the GkS method uses the Fourier transform to represent the spatial derivatives, it is much more accurate than the FD method. Numerical examples show that the GkS method with only four grids per wavelength can achieve the same accuracy as the FD method with 16 grids per wavelength.