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

TL;DR

The paper introduces a fast spectral algorithm for evaluating hyperbolic potentials in 2D, significantly reducing computational complexity for high-frequency wave problems with many sources.

Contribution

It develops a quasi-linear scaling method combining windowed Fourier projection and sum-of-exponentials approximation for efficient wave evaluation.

Findings

Achieves five orders of magnitude speedup over direct methods.

Handles up to a million sources with high accuracy.

Demonstrates effectiveness in large-scale 2D wave simulations.

Abstract

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 $O(M^2N_t^2)$, 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 (c) 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 (c) 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 (c) -- 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.