The PPPM/MPE method, Particle-Particle Particle-Mesh/Multipole Expansion, by Shimada
et al. [46,47] is basically an extension of Hockney and Eastwood's
method [31] for a periodic system with the aid of multipole expansions.
Firstly, the method starts by partitioning the
simulation box into 
cells. At the center of each cell is a mesh point
at which the potential, multipole expansion, etc.
are considered. The electrostatic potential (force) exerted on particle i is decomposed into
the PP (particle-particle) and PM (particle-mesh) interactions.
Secondly, the short-range PP interactions are computed directly between particle
pairs in the same or neighboring cells.
Thirdly, the long-range PM interactions of the remaining cells that are well separated from
particle i are evaluated at i's cell center by expressing the potential due to all
remote cells as a multipole expansion. The PM techniques are then employed [31]
which relay on the use of FFT rather than hierarchical schemes of [26].
The PM potential (force) evaluations are considered to be a smooth function
of the grid coordinates and hence the results can be interpolated back to the particles' locations.
The performance of the P
M/MPE method improved when the twin-range procedure was incorporated in
the following fashion. The PP interactions were calculated at each time step using the most up-to-date
particles' locations, while the PM interactions were only updated every 10-20 time steps. This
improvement has reportedly reduced the CPU time by a factor of three on average.
In their paper [46], the authors stated that PPPM methods alone ``do not give
extremely accurate results'' and hence should be used with caution for precise, long
time-scale simulations.
The paper also compared the accuracy and CPU time of their method
to Hockney and Eastwood's method by examining two systems (BPTI and a random configuration). The comparison
regarded both Hockney's et al. and Shimada's et al. methods as ``nearly comparable'' in overall performance.