Truncated kernel windowed Fourier projection: a fast algorithm for the 3D free-space wave equation

Abstract

We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time steps of size $\Delta t$, a naive potential evaluation at $M$ targets on the same time grid requires $\mathcal O(M^2 N_t)$ work. Our scheme requires $\mathcal{O}\left((M + N^3\log N)N_t\right)$ work, where $N$ scales as $\mathcal O(1/\Delta t)$, i.e., the maximum signal frequency. This is achieved by using the recently-proposed windowed Fourier projection (WFP) method to split the potential into a local part, evaluated directly, plus a smooth history part approximated by an $N^3$-point equispaced discretization of the Fourier transform, where each Fourier coefficient obeys a simple recursion relation. The growing oscillations in the spectral representation (which would be present with a naive use of the Fourier transform) are controlled by spatially truncating the hyperbolic Green's function itself. Thus, the method avoids the need for absorbing boundary conditions. We demonstrate the performance of our algorithm with up to a million sources and targets at 6-digit accuracy. We believe it can serve as a key component in addressing time-domain wave equation scattering problems.