for inventing the Cloud Chamber and made a major contribution to our present
understanding of lightning. Lightning research continued at a steady pace until the
late 1960s when the research became particularly active. This increased interest
was motivated by
The danger of lightning to aircraft or spacecraft,
Solid-state electronics used in computers and other devices,
Improved measurement and observational capabilities due to advancing
technology.
Most lightning research is done by physicists, chemists, meteorologists, and
electrical engineers. Hager [5, 6, 7] was the first mathematician using Maxwell's
equations to develop a three-dimensional mathematical electrical model to sim-
ulate a lightning discharge. His discharge model [6] was obtained by discretizing
Maxwell's equations to obtain a relation between the potential field and current
density due to the motion of charged particles moving under the influence of wind.
Spatial derivatives in his equation were approximated by using volume elements in
space, while the temporal derivatives were estimated by a backward Euler scheme
in time. Since conductivity is very large in the region where the electric field
reaches the breakdown threshold, he evaluated the solution limit as the conductiv-
ity tends to infinity in the breakdown region. In his model [6], the output was the
electric field as a function of time, and the inputs were currents generated by the
flow of charged particles within the thundercloud under the influence of wind.
This dissertation is based on Hager's mathematical model. Some improve-
ments are made compared to Hager's earlier work. For example,
The equations are discretized in a different way allowing us to solve them
by fast fourier transform (FFT) rather than by Cholesky factorization.
Consequently, finer meshes can be employed and larger domain can be
modeled. For example, numerical results reported by Hager et al. [7], the top