  We provide a mathematically proven parallelization scheme for particle
methods on distributed-memory computer systems. Particle methods are a
versatile and widely used class of algorithms for computer simulations and
numerical predictions in various applications, ranging from continuum fluid
dynamics and granular flows, using methods such as Smoothed Particle
Hydrodynamics (SPH) and Discrete Element Methods (DEM) to Molecular Dynamics
(MD) simulations in molecular modeling. Particle methods naturally lend
themselves to implementation on parallel-computing hardware. So far, however, a
mathematical proof of correctness and equivalence to sequential implementations
was only available for shared-memory parallelism. Here, we leverage a formal
definition of the algorithmic class of particle methods to provide a proven
parallelization scheme for distributed-memory computers. We prove that these
parallelized particle methods on distributed memory computers are formally
equivalent to their sequential counterpart for a well-defined class of particle
methods. Notably, the here analyzed parallelization scheme is well-known and
commonly used. Our analysis is, therefore, of immediate practical relevance to
existing and new parallel software implementations of particle methods and
places them on solid theoretical grounds.
