In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (far-field) solution for time-dependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the near-field solution, are imposed as Cauchy data for the radial wave equation in the far-field. Similar to the far-field computation scheme used by Wright, the radial modes in the exterior region are computed using an explicit finite difference solver. However, instead of using an 'infinite grid', we truncate the exterior radial grid at the far-field point of interest, and for each harmonic, impose the same exact NRBC used for the near-field truncation boundary, here expressed in modal form. Using this approach, two different methods for extrapolating the near- field solution to the far-field are possible. In the first, the near-field solution is computed using the exact NRBC, then, based on the solution for the radial modes evaluated on the artificial boundary, the exterior solution may be computed as a post-process. In the second, we show how to compute the far-field solution concurrently with the near-field solution and the NRBC. Numerical studies demonstrate that the method is highly accurate and efficient for direct time-domain computations of far-field solutions.
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