Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.07170

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

Nour G. Al Hassanieh·Leslie Greengard·Alex H. Barnett
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nour G. Al Hassanieh, Leslie Greengard, Alex H. Barnett. 2026-04-08. A spectral method for the rapid evaluation of hyperbolic potentials in two dimensions using windowed Fourier projection. https://arxiv.org/abs/2604.07170

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗