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