Numerical Methods of Electromagnetic Phenomena
in Complex Inhomogeneous Systems
Dr. Wei Cai
Department of Mathematics
University of Noth Carolina at Charlotte
Abstract
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 quantum
transport in nano-electronics. In this talk, we will present two new
algorithm developments for 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 material?s 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 inhomogeneous waveguides under the
paraxial approximation, we need to solve time dependent Schr?dinger
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.