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