Non-standard FDTD for elastic wave simulation: Two-dimensional P-SV case

A scheme for the non-standard finite-difference method in the time-domain (NS-FDTD), 2-Box scheme, is proposed for elastic wave simulations in two dimensions (P-SV). The method improves the accuracy and efficiently reduces grid dispersion and anisotropy. The proposed non-standard scheme is based on two main operations. The first operation replaces spatial grid spacing and time step by their frequency optimized counterparts, called the correction functions, and the second operation introduces an optimum grid stencil for the finite-difference operator of the 2-D Laplacian. The optimal stencil is obtained by introducing two optimization parameters estimated for a design frequency. Error analysis of the proposed scheme (2-Box scheme) shows that specifying the maximum frequency as the design frequency leads to a significant reduction of the grid dispersion over a wide frequency band. We derive the formulations of grid dispersion and stability condition for the scheme. The grid dispersion is investigated, and it is shown that the proposed scheme reduces not only the grid dispersion but also the grid anisotropy significantly. The grid dispersion is insensitive to the Poisson's ratio and size of the time step within the stability limit. Since the wide spatial stencil of the 2-Box scheme might become difficult to implement at the computational domain boundaries, two additional non-standard schemes - 1-Box and 0-Box schemes - are also introduced. The 1-Box scheme uses narrower stencil than the 2-Box scheme, and the 0-Box scheme uses the same stencil as the standard FDTD. Numerical experiments of elastic wave propagation demonstrate the significant superiority of the proposed non-standard schemes over the commonly used standard one. With six grid spacings per minimum wavelength, the 2-Box and 1-Box schemes represent excellent results and the 0-Box scheme has higher accuracy than the standard scheme with seven grid spacings per minimum wavelength.