A fast algorithm for the wave equation using time-windowed Fourier projection

We introduce a new arbitrarily high-order method for the rapid evaluation of hyperbolic potentials (space-time integrals involving the Green's function for the scalar wave equation). With $M$ points in the spatial discretization and $N_t$ time steps of size $\Delta t$, a naive implementation would require $\mathcal O(M^2N_t^2)$ work in dimensions where the weak Huygens' principle applies. We avoid this all-to-all interaction using a smoothly windowed decomposition into a local part, treated directly, plus a history part, approximated by a $N_F$-term Fourier series. In one dimension, our method requires $\mathcal O\left((M + N_F \log N_F)N_t\right)$ work, with $N_F =\mathcal O(1/\Delta t)$, by exploiting the non-uniform fast Fourier transform. We demonstrate the method's performance for time-domain scattering problems involving a large number $M$ of springs (point scatterers) attached to a vibrating string at arbitrary locations, with either periodic or free-space boundary conditions. We typically achieve 10-digit accuracy, and include tests for $M$ up to a million.