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

TL;DR

This paper presents a high-order, fast algorithm for evaluating hyperbolic potentials related to the wave equation, significantly reducing computational complexity using a Fourier series approach and non-uniform FFT, enabling large-scale time-domain scattering simulations.

Contribution

The paper introduces a novel high-order method that reduces the computational cost of wave equation evaluations from quadratic to near-linear in the number of spatial points and time steps, using a Fourier series decomposition and non-uniform FFT.

Findings

Achieves 10-digit accuracy in simulations.

Handles up to one million scatterers efficiently.

Reduces computational complexity to near-linear in problem size.

Abstract

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.