The thesis considers and examines methods of surface propagation,
where the normal velocity of the surface depends on the local curvature
and the gradient of the surface. Such fronts occur in many different physical
situations from the growth of crystals to the spreading of flames. A
number of different methods are considered to find solutions to these
physical problems. First the motion is modelled by partial differential
equations and numerical methods are developed for solving these equations.
The numerical methods involve characteristic, finite differences
and transformation of the equations. Stability of the solutions is also
briefly considered. Secondly the fronts are modelled by using a cellular
approach which subdivides space into regions of small cells. The fronts are
assumed to propagate through the region according to stochastic rules.
Monte-Carlo simulations are carried out using this approach. Results of
the simulations are carried out in two-dimensions and three-dimensions
for a number of interesting physical examples.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.