New versions of image approximations to the ionic solvent induced reaction field

A recent article by Deng and Cai (Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: High-order image approximations for reaction fields, to appear in J. Comput. Phys.) introduced two fourth-order image approximations to the reaction field for a charge inside a dielectric sphere immersed in a solvent of low ionic strength. To represent such a reaction field, the image approximations employ a point charge at the classical Kelvin image point and two line charges that extend from this Kelvin image point along the radial direction to infinity, with one decaying to zero and the other growing to infinity. In this paper, alternative versions of the fourth-order image approximations are presented, using the same point charge but three different line charges, all decaying to zero along the radial direction. Similar discussions on how to approximate the line charges by discrete image charges and how to apply the resulting multiple discrete image approximations together with the fast multipole method are also included.