The mean field plus approximation

A new algorithm for approximating the energy and fields between clusters of spatially distinct particles will be described. The case where the particles are magnetic grains will be the focus, but the method is more broadly applicable. This approximation accomplishes the same task as multipole expansions for computing cluster-cluster interactions, but it seems to be both faster and more accurate. It can be viewed as a generalization of a certain mean field approximation and so will be called the mean field plus approximation. The lowest-order version of the approximation will be described first and then a systematic way to improve the approximation will be given. Results of numerical experiments comparing the computational time and accuracy for the mean field approximation, the multipole expansion and the mean field plus approximation will be given.