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The rapid evaluation of potential fields in particle systems. (English) Zbl 1001.31500
ACM Distinguished Dissertations. Cambridge, MA: The MIT Press. xiv, 91 p. (1988).
This thesis presents new and remarkably effective procedures for the computation of potentials and forces in systems of many charged particles. The innovative stratagem central to the author’s treatment is nevertheless easily apprehended.
Consider a cluster of $$M$$ charged particles and, in some disjoint region of space, a cluster of $$N$$ charged particles. An evaluation of the total electrostatic interaction energy between the two clusters would appear, at first sight, to require a computational effort proportional to $$MN$$. However, by virtue of the amenable analytic properties of the Coulomb potential it is possible instead to express the interaction energy in terms of products of multipole moments of the two clusters. The computation of a multipole moment involves a computational effort proportional either to $$M$$, or to $$N$$, according to whichever cluster is pertinent. The number of multipoles needed depends upon the overall precision required, but is finite and independent of $$M$$ and $$N$$, and independent also of the detailed dispositions of the particles within their respective clusters. The total computational effort thus becomes proportional to $$M+N$$ rather than to $$MN$$, resulting in enormous savings if $$M$$ and $$N$$ are large.
The thesis provides, on the theoretical side, a full account of the multipole formalism and the associated truncation errors in both two and three dimensions, along with some new theorems expressing the transformation of multipole moments under translation of the origin. On the computational side, it sets out strategies for the economical partitioning of the whole system into disjoint clusters, with full algorithms for computer implementation, including numerical methods for the imposition of various boundary conditions.
Several specimen computations performed in two dimensions with up to $$100\times 2^8$$ particles are reported. They provide substantive evidence of the power and accuracy of the multipole method in large systems.
Possible applications in astronomy, physics, chemical physics, fluid dynamics and numerical complex analysis are succinctly outlined.

##### MSC:
 31C20 Discrete potential theory 31-04 Software, source code, etc. for problems pertaining to potential theory 70-08 Computational methods for problems pertaining to mechanics of particles and systems 78-04 Software, source code, etc. for problems pertaining to optics and electromagnetic theory 78A30 Electro- and magnetostatics