Editing Discrete element method (section)

==Long-range forces==

When long-range forces (typically gravity or the Coulomb force) are taken into account, then the interaction between each pair of particles needs to be computed. Both the number of interactions and cost of computation [[quadratic growth|increase quadratically]] with the number of particles. This is not acceptable for simulations with large number of particles. A possible way to avoid this problem is to combine some particles, which are far away from the particle under consideration, into one pseudoparticle. Consider as an example the interaction between a star and a distant [[galaxy]]: The error arising from combining all the stars in the distant galaxy into one point mass is negligible. So-called tree algorithms are used to decide which particles can be combined into one [[pseudoparticle]]. These algorithms arrange all particles in a tree, a [[quadtree]] in the two-dimensional case and an [[octree]] in the [[Three-dimensional space|three-dimensional]] case.

However, simulations in molecular dynamics divide the space in which the simulation take place into cells. Particles leaving through one side of a cell are simply inserted at the other side (periodic [[boundary condition]]s); the same goes for the forces. The force is no longer taken into account after the so-called cut-off distance (usually half the length of a cell), so that a particle is not influenced by the mirror image of the same particle in the other side of the cell. One can now increase the number of particles by simply copying the cells.

Algorithms to deal with long-range force include:
* [[Barnes–Hut simulation]],
* the [[fast multipole method]].