A Unifying Mathematical Definition of Particle Methods

Particle methods are a widely used class of algorithms for computer simulation of complex phenomena in various fields, such as fluid dynamics, plasma physics, molecular chemistry, and granular flows, using diverse simulation methods, including Smoothed Particle Hydrodynamics (SPH), Particle-in-Cell (PIC) methods, Molecular Dynamics (MD), and Discrete Element Methods (DEM). Despite the increasing use of particle methods driven by improved computing performance, the relation between these algorithms remains formally unclear, and a unifying formal definition of particle methods is lacking. Here, we present a rigorous mathematical definition of particle methods and demonstrate its importance by applying it to various canonical and non-canonical algorithms, using it to prove a theorem about multi-core parallelizability, and designing a principled scientific computing software based on it. We anticipate that our formal definition will facilitate the solution of complex computational problems and the implementation of understandable and maintainable software frameworks for computer simulation.