Optimally accurate second-order time-domain finite difference scheme for the elastic equation of motion: one-dimensional case

We previously derived a general criterion for optimally accurate numerical operators for the calculation of synthetic seismograms in the frequency domain (Geller & Takeuchi 1995). We then derived modified operators for the Direct Solution Method (DSM) (Geller & Ohminato 1994) which satisfy this general criterion, thereby yielding significantly more accurate synthetics (for any given numerical grid spacing) without increasing the computational requirements (Cummins 1994; Takeuchi, Geller & Cummins 1996; Cummins, Takeuchi & Geller 1997). In this paper, we derive optimally accurate time-domain finite difference (FD) operators which are second order in space and time using a similar approach. As our FD operators are local, our algorithm is well suited to massively parallel computers. Our approach can be extended to other methods (e.g. pseudo-spectral) for solving the elastic equation of motion. It might also be possible to extend this approach to equations other than the elastic equation of motion, including non-linear equations.