Application of Kummer's Transformation to the Efficient Computation of the 3-D Green's Function With 1-D Periodicity

The 3-D homogeneous Green's function with 1-D periodicity is commonly expressed as spatial and spectral infinite series that may show very slow convergence. In this work Kummer's transformation is applied to the spatial series in order to accelerate its convergence. By retaining a sufficiently large number of asymptotic terms in the application of Kummer's transformation, the spatial series is split into a set of series which can be accurately obtained with very low computational effort. The numerical results obtained show that, when the number of asymptotic terms retained in Kummer's transformation is large enough, the convergence acceleration method proposed in this work is always faster than existing acceleration methods such as the spectral Kummer-Poisson's method and Ewald's method.