Journal of Scientific Computing

, Volume 79, Issue 2, pp 827–866 | Cite as

Particle-without-Particle: A Practical Pseudospectral Collocation Method for Linear Partial Differential Equations with Distributional Sources

  • Marius OlteanEmail author
  • Carlos F. Sopuerta
  • Alessandro D. A. M. Spallicci


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.


Pseudospectral methods Distributionally-sourced PDEs Gravitational self-force Neural populations Price formation 



Due by MO to the Natural Sciences and Engineering Research Council of Canada, by MO and ADAMS to LISA France-CNES, and by MO and CFS to the Ministry of Economy and Competitivity of Spain, MINECO, Contracts ESP2013-47637-P, ESP2015-67234-P and ESP2017-90084-P.


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Authors and Affiliations

  1. 1.Institute of Space Sciences (ICE, CSIC)Cerdanyola del Vallès (Barcelona)Spain
  2. 2.Institut d’Estudis Espacials de Catalunya (IEEC)BarcelonaSpain
  3. 3.Departament de Física, Facultat de CiènciesUniversitat Autònoma de BarcelonaCerdanyola del Vallès (Barcelona)Spain
  4. 4.Observatoire des Sciences de l’Univers en région Centre (OSUC)Université d’OrléansOrléansFrance
  5. 5.Laboratoire de Physique et Chimie de l’Environnement et de l’Espace (LPC2E)Centre National de la Recherche Scientifique (CNRS)OrléansFrance
  6. 6.Pôle de Physique, Collegium Sciences et Techniques (CoST)Université d’OrléansOrléansFrance
  7. 7.Departamento de Astrofísica, Cosmologia e Interações Fundamentais (COSMO)Centro Brasileiro de Pesquisas Físicas (CBPF)Urca, Rio de JaneiroBrazil

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