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

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