Particle–Particle–Particle–Mesh (P³M) is a Fourier-based

Ewald summation Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crys ...

method to calculate potentials in

N-body simulation In physics and astronomy, an ''N''-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity (see n-body problem, ''n''-body problem for other applications). ''N''-body ...

s. The potential could be the electrostatic potential among N point charges i.e.

molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics ( ...

, the gravitational potential among N gas particles in e.g. smoothed particle hydrodynamics, or any other useful function. It is based on the particle mesh method, where particles are interpolated onto a grid, and the potential is solved for this grid (e.g. by solving the discrete

Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...

). 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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Further reading