In plasma simulations, where the speed of light divided by a characteristic
length is at a much higher frequency than other relevant parameters in the
underlying system, such as the plasma frequency, implicit methods begin to play
an important role in generating efficient solutions in these multi-scale
problems. Under conditions of scale separation, one can rescale Maxwell's
equations in such a way as to give a magneto static limit known as the Darwin
approximation of electromagnetics. In this work, we present a new approach to
solve Maxwell's equations based on a Method of Lines Transpose (MOLT)
formulation, combined with a fast summation method with computational
complexity O(NlogN), where N is the number of grid points (particles).
Under appropriate scaling, we show that the proposed schemes result in
asymptotic preserving methods that can recover the Darwin limit of
electrodynamics