An improved transformation and optimized sampling scheme for the numerical evaluation of singular and near-singular potentials

In this work we compare four transformations for numerically computing singular and near-sinuglar integrals. These schemes employ singularity cancellation methods in which a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. Recently, a new singularity cancellation scheme, the arcsinh transformation, was presented for handling 1/R singularities. For singular integrals the method not only has several advantages over singularity subtraction methods, but also improves on some aspects of other singularity cancellation methods such as polar and Duffy transformations. One drawback of the scheme, however, is its inability to efficiently calculate near-singular integrals. To this end, we summarize and compare the performance of four transformations to handle near-singularities: the arcsinh transform, Duffy and polar transformations extended to treat near-singularities, and, a new scheme referred to as the radial-angular transformation. We find that the performance of the radial-angular transformation is much better than the other transformations for the near-singular case and comparable to them for the singular case. Subsequently, we describe a method for optimizing computation of the radial-angular transformation when a required accuracy is specified.