Abstract: Wave propagation in unbounded domains arises from diverse applications. Intensive research has been devoted to frequency-domain simulation, e.g., the time-harmonic Helmholtz problems and Maxwell's equations. Here, we are interested in time-domain computation, which is known to be more flexible in capturing wide-band signals and modeling more general material inhomogeneity and nonlinearity. In this talk, we shall show how to use tools in complex analysis to analytically evaluate circular and spherical nonreflecting boundary conditions (NRBCs), and how to efficiently deal with time-space globalness of such boundary conditions. Fast spectral-Galerkin solvers together with stable time integration and techniques for handling general irregular scatterers will be introduced for the simulation. We intend to demonstrate that the interplay between analytic tools and accurate numerical means can lead to efficient methodologies for challenging simulations.