Theoretical basis for numerically exact three-dimensional time-domain algorithms

In a one-dimensional (1D) homogeneous space, the classic Yee finite-difference timedomain (FDTD) algorithm is numerically exact when operated at the Courant stability limit. Numerically exact is taken to mean that, to within the sampling limit imposed by the discretization in space and time, the only errors are due to the finite precision of digital computer arithmetic. Unfortunately, the Yee algorithm is not numerically exact in two or more dimensions. However, using the design shown here, three-dimensional (3D) spatial differential operators can have 1D dispersion properties. Just as the space and time errors can be made to cancel in the 1D Yee algorithm, 3D algorithms (for hyperbolic systems of coupled first order equations) in an unbounded homogeneous space can be constructed which are, in theory, numerically exact. The differential operators presented here extend over a localized non-zero volume, unlike the usual nabla (or Del) operator which acts at a point. Our computer implementations are based on reconstruction methods, producing global range operators, thus our implementations of these operators are computationally expensive. A sample implementation of an approximate electromagnetic algorithms is described and is shown to produce results that are superior to the classic Yee algorithm for the cubic resonator problem.