  We review some recent developments in numerical algorithms to solve the
time-dependent Maxwell equations for systems with spatially varying
permittivity and permeability. We show that the Suzuki product-formula approach
can be used to construct a family of unconditionally stable algorithms, the
conventional Yee algorithm, and two new variants of the Yee algorithm that do
not require the use of the staggered-in-time grid. We also consider a one-step
algorithm, based on the Chebyshev polynomial expansion, and compare the
computational efficiency of the one-step, the Yee-type, the
alternating-direction-implicit, and the unconditionally stable algorithms. For
applications where the long-time behavior is of main interest, we find that the
one-step algorithm may be orders of magnitude more efficient than present
multiple time-step, finite-difference time-domain algorithms.
