Long Range Forces

There are a couple of approaches here.

• PM. One (not so good) is a particle-mesh method. Essentially one converts particle quantities (like a mass in a gravitational problem) to a field quantity (like density). Then, solve the field eqns on a mesh, interpolate the resulting potential to particle locations, change the field into a force, and move the particle with that force.
• PPPM. Better is a particle-particle particle-mesh hybrid. Nearby interactions are handled directly (PP), while more distant interactions are captured through a PM approach.
• PM2 and NGPM. Particle Multiple Mesh, and Nested Grid Particle Mesh are variants of PM and PPPM methods, with adaptativity in meshes included.
• Fast Multipole Methods of Rokhlin and Greengard. Starting with a multipole expansion and a heirarchy of regular, nested grids, an O(N log(N)) algorithm is relatively straightforward to derive. FMM uses asymptotic techniques to control errors, translates the multipole expansion to cell centers, and uses symmetry in the problem to account for interactions among parent and child grids and distant particles in an optimal O(N) fashion. See www.icsi.berkeley.edu/cs267/lecture27/lecture27.html.