Fast and accurate computation of electromagnetic phenomena plays an important role in understanding the underlying physics for many complex physical and biological systems, such as lasing in optical fiber lasers, electrostatics forces in solvation model of biomolecules, and irradiation damage in materials under extreme conditions. In this talk, we will present two new algorithm developments with applications in these areas.

Image Charge Approximations of Reaction Fields and FMM for Charges inside a Dielectric Sphere

The reaction field of a charge inside a dielectric sphere, induced by a surrounding dissimilar dielectric medium, has applications in the study of electrostatic forces in the defect evolutions in material under extreme neutron irradiation, and hybrid explicit/implicit solvation models for biomolecules. In both cases, the long range Coulomb interactions have been identified as of primary influence in materials resistance to amorphorization under extreme conditions in the first case, and the free energy and the solvation study of biomolecules for the second. We have developed new discrete image charge approximations for the reaction field of a charge inside a dielectric sphere at high accuracy with only 2-3 image charges. Based on this result, we have extended the Fast Multipole Method to calculate the electrostatic interactions of charges inside or outside a dielectric sphere. The resulting O(N) algorithm has applications in computational materials and biology.

A Generalized Discontinuous Galerkin (GDG) Method based on Split Distributions for PDE with Nonsmooth Solutions

To model optical wave propagations in inhomogenous waveguides under the paraxial approximation, we need to solve time dependent Schrödinger equations with nonsmooth solutions as a result of field discontinuities at material interfaces. We will present a new type of discontinuous Galerkin method based on split distributions and their incorporations into the PDEs to account for jumps in solutions and derivatives. Special integration by parts formula for the split distributions is developed. The resulting generalized discontinuous Galerkin (GDG) method will be flexible to handle various types of interface jump conditions (time dependent and nonlinear) with high accuracy and easy to extend to multi-dimensional and other type PDEs with nonsmooth solutions. A full vector GDG-BPM (beam propagation method) will be developed to study gain guided fiber laser for efficient generations of high energy power sources.