Higher-Order Fdtd Methods For Large Problems

The Finite-Diierence Time-Domain (FDTD) algorithm provides a simple and eecient means of solving Maxwell's equations for a wide variety of problems. In Yee's uniform grid FDTD algorithm the derivatives in Maxwell's curl equations are replaced by central diierence approximations. Unfortunately, numerical dispersion and grid anisotropy are inherent to FDTD methods. For large computational domains, e.g., ones that have at least one dimension forty wavelengths or larger, phase errors from dispersion and grid anisotropy in the Yee algorithm (YA) can be signiicant unless a small spatial discretization is used. For such problems, the amount of data that must be stored and calculated at each iteration can lead to prohibitive memory requirements and high computational cost. To decrease the expense of FDTD simulations for large scattering problems two higher-order methods have been derived and are reported here. One method is second-order in time and fourth-order in space (2-4); the other is second-order in time and sixth-order in space (2-6). Both methods decrease grid anisotropy and have less dispersion than the YA at a set discretization. Also, both permit a coarser discretization than the YA for a given error bound. To compare the accuracy of the YA and higher-order methods both transient and CW simulations have been performed at a set discretization. In general, it has been found that the higher-order methods are more accurate than the YA due to the reduced grid anisotropy and dispersion. However, the higher-order methods are not as accurate as the YA for the simulation of surface waves. This is attributed to the larger spatial stencil used in calculating the elds for the higher-order methods. More research is needed to examine the accuracy of higher-order methods at material boundaries.