A spectral method for the rapid evaluation of hyperbolic potentials in two dimensions using windowed Fourier projection

We present a fast algorithm for evaluating the (non-smooth) solution of the free-space two-dimensional (2D) scalar wave equation with many point sources, each with a high-frequency band-limited time signature. Such an algorithm is key to an efficient time-domain scattering solver using spatially-discretized hyperbolic layer potentials. Given $M$ sources/targets and $N_t$ time steps, direct evaluation costs $O(M^2{N}_t^2)$, due to the history dependence. We develop a quasi-linear scaling algorithm that splits the solution at a given time into (a) a non-smooth time-local part, (b) a (smooth) near history involving sources up to $\mathcal{O}(1)$ domain traversal times into the past, plus © a (very smooth) far history comprising all waves emitted before the near history. The local part is computed directly via high-order quadrature. A naive spatial Fourier transform for (b) plus © would be both slowly converging and arbitrarily oscillatory as time progresses. Yet in (b) the oscillations are controlled, so we use the recent truncated windowed Fourier projection (TK-WFP) method to give rapid convergence. For © – present due to the weak Huygens’ principle – we exploit a new large-time sum-of-exponentials approximation of the free-space wave kernel. Numerical examples with up to a million sources and targets, a domain of $300\times 300$ wavelengths, and 6-digit accuracy, show an acceleration of five orders of magnitude relative to direct evaluation.