A Generalized Discontinuous Galerkin Method

Kai Fan holds a B.Sc (2003) in Mathematics from Hong Kong Baptist
University, China,
and a
Ph.D (2008) in Applied Mathematics from the University of North
Carolina at Charlotte,
US. In 2008, he joined North Carolina State University as a
research associate.

In this book, a new Generalized Discontinuous
Galerkin (GDG)
method for Schrodinger equations with nonsmooth
solutions is
proposed. The numerical method is based on a
reformulation of
Schrodinger equations, using split distributional
variables and their
related integration by parts formulae to account for
solution jumps
across material interfaces. GDG can handle time
dependent and
general nonlinear jump conditions. And numerical
results validate
the high order accuracy and the flexibility of the
method for various
types of interface conditions. As one of GDG's
application, a new
vectorial generalized discontinuous Galerkin beam
propagation
method (GDG-BPM) for wave propagations in inhomogeneous
optical waveguides is also included. The resulting
GDG-BPM takes
on four formulations for either electric or magnetic
field. GDG-
BPM's unique feature of handling interface jump
conditions and its
flexibility in modeling wave propagations in
inhomogeneous optical
fibers is shown by various numerical results.