  This paper presents the generalized formulations of fundamental schemes for
efficient unconditionally stable implicit finite-difference time-domain (FDTD)
methods. The fundamental schemes constitute a family of implicit schemes that
feature similar fundamental updating structures, which are in simplest forms
with most efficient right-hand sides. The formulations of fundamental schemes
are presented in terms of generalized matrix operator equations pertaining to
some classical splitting formulae, including those of alternating direction
implicit, locally one-dimensional and split-step schemes. To provide further
insights into the implications and significance of fundamental schemes, the
analyses are also extended to many other schemes with distinctive splitting
formulae. Detailed algorithms are described for new efficient implementations
of the unconditionally stable implicit FDTD methods based on the fundamental
schemes. A comparative study of various implicit schemes in their original and
new implementations is carried out, which includes comparisons of their
computation costs and efficiency gains.
