An influence matrix particle – particle particle - mesh algorithm with exact particle – particle correction

The classical N -body problem consisting of the interaction of N computational particles, is the key computational issue in a number of diverse fields of science including chemistry, fluid dynamics, and astrophysics. For particles interacting through long range potentials the direct evaluation of their mutual interaction nominally scales as OðN Þ. In order to reduce this computational cost a simple truncation of the interaction potential appears to be a viable approach for homogeneous (neutral) systems [1], reducing the operational count to OðNÞ. While this approximation may suffice for the modelling of the electrostatic interaction in molecular dynamics (MD) simulations, the assumption is insufficient for problems in fluid dynamics and astrophysics. In these cases, and for MD simulations requiring higher accuracy than warranted by a simple truncation, fast methods have been devised such as the Barnes–Hut [3] and the fast multipole method [7], as well as hybrid mesh based algorithms including the particle-mesh (PM) and the particle–particle particle-mesh (PPPM) algorithm originally proposed by Hockney and Eastwood [9]. While fast multipole methods offer an operational cost of OðNÞ and an exact enforcement of the freespace boundary condition, hybrid particle-mesh algorithms with an operational cost of OðN logNÞ or OðNÞ are often found to be computationally superior for problems in simple geometries and for periodic systems. The particle-mesh algorithm is based on the observation that the three-dimensional, free-space Green s function to the Laplace operator (r2) is 1=4pr, and they attain their efficiency by employing fast Fourier transforms or fast iterative solvers for the solution of the field equation on a mesh. The density field is constructed on the mesh from the strength of the particles using a smooth projection, which also serves to interpolate the computed force field back onto the particles. For particle systems involving non-smooth density fields the exact force field will contain sub-grid scales not resolved by the PM algorithm, and an explicit particle–particle (PP) correction term (f) is required to Journal of Computational Physics 184 (2003) 670–678