Closed form FDTD-compatible Green’s function based on combinatorics

Abstract

The advantages of the finite-difference-time-domain (FDTD) method are often hampered by the need to model large “white spaces” between and around scattering objects. In the continuous realm, these large spaces are customarily bridged by the usage of integral operators that transform the sources to any observation point using an appropriate Green’s function. A companion procedure for the discretized world can be realized in principle by straightforward sampling of the continuous Green’s function. However, such a procedure does not track the FDTD algorithm and hence yields different results. Alternatively, an FDTD-compatible discrete Green’s function is derived in this work with the Yee-discretized Maxwell’s equations as first principles. The derivation involves a process of counting many combinations of paths in the spatial–temporal grid leading to recursive combinatorial expressions that are solved in closed form. Numerical implementations of the resultant Green’s function in short-pulse propagation problems produce results validated by conventional FDTD computations. The advantages of efficient computations over large distances, in particular with regard to short pulses, are thus demonstrated.