P3M

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{{for|the aircraft|Martin P3M}}

Particle–Particle–Particle–Mesh (P³M) is a Fourier-based Ewald summation method{{cite web|url =http://people.ee.duke.edu/~ayt/ewaldpaper/node15.html#SECTION00041000000000000000|archive-url =https://web.archive.org/web/20090821085337/http://people.ee.duke.edu/~ayt/ewaldpaper/node15.html#SECTION00041000000000000000|url-status =dead|archive-date =2009-08-21|title =Fourier-based Ewald Summation Methods (Web Version)|accessdate =2009-03-01}} {{cite journal|doi =10.1016/0010-4655(96)00016-1|title=Fourier-based Ewald Summation Methods (Published Version) |bibcode = 1996CoPhC..95...73T |last1=Toukmaji |first1=Abdulnour Y. |last2=Board |first2=John A. |journal=Computer Physics Communications |year=1996 |volume=95 |issue=2 |page=73 }} to calculate potentials in N-body simulations.{{cite journal|doi =10.1016/S0010-4655(84)82783-6|title =P3M3DP-the three-dimensional periodic particle-particle/particle-mesh program|year =1984|bibcode = 1984CoPhC..35..618E |last1 =Eastwood|first1 =J.W.|last2 =Hockney|first2 =R.W.|last3 =Lawrence|first3 =D.N.|journal =Computer Physics Communications|volume =35}}{{cite web|url = http://link.aip.org/link/?JCPSA6/109/7694/1|title = How to mesh up Ewald sums. II. An accurate error estimate for the particle–particle–particle-mesh algorithm |accessdate = 2009-03-01}}{{cite journal|title = N-body simulations, section P3M and PM Tree Codes| journal=Scholarpedia | date=20 May 2008 | volume=3 | issue=5 | page=3930 | doi=10.4249/scholarpedia.3930 | doi-access=free | last1=Trenti | first1=Michele | last2=Hut | first2=Piet | bibcode=2008SchpJ...3.3930T }}

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