A novel low-dispersive [2,2] finite difference method

Abstract

The dispersion relation and convergence of a novel (2,2) finite-difference modified (5)-(8) (FDM) scheme which has fourth order convergence and excellent broadband characteristics, are presented. Accuracy of several low-dispersion finite-difference time-domain (FDTD) schemes in 2-D is compared with that of the FDM, via direct evaluation of the dispersion relation. Convergence of the FDTD and the FDM in 2-D are also examined. Index Terms— Finite differences, numerical dispersion, convergence. I. INTRODUCTION Standard numerical techniques, like the finite- difference time-domain (1)-(2), and the finite integra- tion technique (3)-(4) provide the estimates of the error of the discretized numerical operators rather than the error of the numerical solutions computed using these operators (5) - (8). The modified finite- difference operators presented here are optimally ac- curate, which means the numerical solutions com- puted using these modified operators provide the greatest attainable accuracy for a particular type of scheme, e.g., second-order finite differences, for some particular grid spacing. The modified operators lead to an implicit scheme. Using the first-order Born approximation, this implicit scheme is transformed into a two step explicit scheme. The stability condition of the modified scheme is equivalent to that of the corresponding conventional schemes (like FDTD and FIT) (8).