Particle-without-Particle: a practical pseudospectral collocation method for linear partial differential equations with distributional sources

TL;DR

The paper introduces the 'Particle-without-Particle' method, a pseudospectral collocation technique that efficiently solves linear PDEs with distributional sources by avoiding singularities and improving convergence.

Contribution

It presents a novel domain decomposition approach that replaces delta sources with jump conditions, enabling accurate solutions without singular behavior.

Findings

Achieves improved convergence rates over traditional delta approximation methods.

Successfully applied to hyperbolic, parabolic, and elliptic PDEs with distributional sources.

Proves the method's validity for any linear PDE with delta distribution sources.

Abstract

Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular ("particle"-like) source modeling invariably introduces nontrivial computational obstacles. A common method to circumvent these is through some form of delta function approximation procedure on the computational grid; however, this often carries significant limitations on the efficiency of the numerical convergence rates, or sometimes even the resolvability of the problem at all. In this paper, we present an alternative technique for tackling such equations which avoids the singular behavior entirely: the "Particle-without-Particle" method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev-Lobatto) grids such that the "particle" is always at the interface between them; thus, one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. We prove here that this method yields solutions to any linear PDE the source of which is any linear combination of delta distributions and derivatives thereof supported on a one-dimensional subspace of the problem domain. We then implement it to numerically solve a variety of relevant PDEs: hyperbolic (with applications to neuroscience and acoustics), parabolic (with applications to finance), and elliptic. We generically obtain improved convergence rates relative to typical past implementations relying on delta function approximations.