P3M
Encyclopedia
Particle-Particle-Particle Mesh (P3M) is a Fourier-based Ewald summation
Ewald summation
Ewald summation, named after Paul Peter Ewald, is a method for computing the interaction energies of periodic systems , particularly electrostatic energies. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an...

 method to calculate potentials in N-body simulation
N-body simulation
An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...

s.

The potential could be the electrostatic potential among N point charges i.e. molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...

, the gravitational potential among N gas particles in e.g. smoothed particle hydrodynamics
Smoothed particle hydrodynamics
Smoothed-particle hydrodynamics is a computational method used for simulating fluid flows. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography...

, or any other useful function. It is based on the particle mesh
Particle Mesh
Particle Mesh is a computational method for determining the forces in a system of particles. These particles could be atoms, stars, or fluid components and so the method is applicable to many fields, including molecular dynamics and astrophysics. The basic principle is that a system of particles...

 method, where particles are interpolated onto a grid, and the potential is solved for this grid (e.g. by solving the discrete Poisson equation). This interpolation introduces errors in the force calculation, particularly for particles that are close together. Essentially, the particles are forced to have a lower spatial resolution during the force calculation. The P3M algorithm attempts to remedy this by calculating the potential through a direct sum for particles that are close, and through the particle mesh method for particles that are separated by some distance.
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