The plane wave time-domain (PWTD) algorithm permits the efficient evaluation of linear (scalar and vector) wave fields due to bandlimited transient sources. In essence, PWTD schemes constitute extensions of the
frequency domain fast multipole method (Helmholtz equation) to the time domain (wave equation). This presentation will review the state of the art in the field, with a special focus on the development of PWTD kernels for two-dimensional, low-frequency, and quasi-planar applications. Contrary to standard free-space 3D kernels, these kernels require in-level multiscale
decompositions to resolve long Green function tails (2D), window plane wave expansions (quasi-planar expansions), or generate electromagnetic bullets (low frequency fields). The application of these PWTD kernels to the solution of various scattering problems will be demonstrated..
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