Optimally accurate second-order time-domain finite-difference scheme for acoustic, electromagnetic, and elastodynamic wave modeling: one-dimensional case

Numerical methods are extremely useful in solving real-life problems with complex materials and geometries. However, numerical methods in the time domain stiffer from artificial numerical dispersion. Standard numerical techniques which are second-order in space and time, like the conventional finite-difference three point (FD3) method, finite-difference time-domain (FDTD) method, and finite integration technique (FIT), provide estimates of the error of discretized numerical operators rather than the error of the numerical solutions computed using these operators. Here, optimally accurate time-domain (TD) finite-difference (FD) operators which are second-order in time as well as in space are derived. Optimal accuracy means the greatest attainable accuracy for a particular type of scheme, e.g., second-order FD, for some particular grid spacing. The modified FD scheme - FD modified: FDM - presented here attains reduction of numerical dispersion almost by a factor of 40 compared to the FD3, FDTD, and FIT. The CPU time for the FDM scheme is twice of that required by FD3 method. The modified operators lead to an implicit scheme, which is approximated by a predictor-corrector scheme yielding a two step explicit scheme. The possibility of extending this method to a staggered grid approach is also presented. Finally the comparison between analytical solution, FDTD/FIT method, FD3 method and FDM scheme with simulation results is depicted. Further examples are given in the presentation.