An O(NS Nt log² Nt) method for evaluating convolutions with the time domain periodic Green's function

Analysis of periodic structures in time domain is an increasingly important tool in the design of a wide range of novel structures. Time Domain Integral Equation based methods provide an accurate means of solving transient periodic problems, but require costly convolutions that scale as O(N_t²N_s²), where N_t and N_s are the temporal and spatial degrees of freedom. This work proposes a fast method for effecting convolutions with the periodic Greens function that scales as O(N_s N_t log² N_t). The method relies on a temporal Floquet expansion of the periodic Greens function that is accelerated in space using the O(N_s) method of Accelerated Cartesian Expansions and in time using an O(N_t log² N_t) blocked FFT scheme.