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Abstract
A class of finite-difference time-domain (FDTD) schemes is developed, for the solution of Maxwell's equations, that exhibits improved isotropy and dispersion characteristics. This is achieved by improving the two-dimensional Laplacian approximation associated with the curl-curl operator. The development of this method is based on the observation that in a two-dimensional space the Yee-algorithm approximates the aforementioned Laplacian operator via a strongly anisotropic 5-point representation. It is demonstrated that with the aid of a transversely extended curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling, an artifact of staggered-collocated grids. Then it is demonstrated that the above methodology is directly applicable in three-dimensions. The properties of the resulting schemes are analyzed and it is found that they exhibit the same favorable characteristics as their two-dimensional counterparts. Additionally, possible modifications for the extended curl operator are proposed which result in higher order performing schemes. First an alternative extended curl operator is derived based on a 25-point isotropic Laplacian discretization. It is shown that the corresponding scheme is fourth order accurate in space, exhibits isotropy up to sixth order and has a higher Courant number than other candidate schemes. Second, the extended-curl operator is combined with fourth order time derivatives via a modified equation approach. The resulting scheme is sixth order isotropic and exhibits a Courant number that is almost unity. Finally, a rigorous, simple and accurate methodology is described which allows the optimization of the original extended curl scheme for a given grid resolution. Representative numerical simulations are performed that validate the theoretically derived results.
