Application of transforms to accelerate the summation of periodic free-space Green's functions

The results of using two acceleration techniques to accelerate the convergence of free-space Green's functions of the Helmholtz equation are presented. An overview of the acceleration methods is given. The techniques make use of Kummer's Poisson's, and Shanks's transforms. The application of Shanks's transform improves dramatically the convergence of the one-dimensional free-space Green's functions. This is indicated by the computation time, which in some cases is reduced by a factor of 10 over the direct summation of the series. The first-order acceleration with Shanks's transform converges faster than first-order acceleration in each case. The advantage offered by the use of Shanks's transform is that no analytical work has to be done to the series. Numerical results include relative error versus number of terms taken in the series, and computation time versus a predefined convergence factor.>