One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation

Authors:
C. M. Linton;I. Thompson
Affiliations:
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK;Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK
Venue:
Journal of Computational Physics
Year:
2009

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Abstract

The accurate and efficient computation of lattice sums for the three-dimensional Helmholtz equation is considered for the cases where the underlying lattice is one- or two-dimensional. We demonstrate, using careful numerical computations, that the reduction method, in which the sums for a two-dimensional lattice are expressed as a sum of one-dimensional lattice sums leads to an order-of-magnitude improvement in performance over the well-known Ewald method. In the process we clarify and improve on a number of results originally formulated by Twersky in the 1970s.