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The rapid evaluation of potential fields in three dimensions. (English) Zbl 0661.70006
Vortex methods, Proc. UCLA Workshop, Los Angeles/Cal. 1987, Lect. Notes Math. 1360, 121-141 (1988).
[For the entire collection see Zbl 0648.00012.]
For large-scale three-dimensional particle interaction problems a fast numerical method is described to evaluate Coulomb fields or gravitational fields. The fast multiple algorithm is based on a box-refinement which subdivides a given cube into smaller ones. Clustering of \(N\) particles in the box is taken into account and multiple expansions for the clusters are advantageously exploited for the calculation of the force fields. This leads to an enormous reduction of the computational complexity of the evaluation of the potential fields. For Legendre polynomials two new useful addition theorems are proved. The storage requirement of the algorithm is \(O(N)\). By symmetry arguments for the coefficients of the shifted expansions a remarkable reduction of work is gained. These results are very valuable and useful for all problems arising in mechanics and fluid mechanics involving a large number of particles.
Reviewer: K.G.Roesner

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