General n-point formulae for difference operators and their errors are derived in terms of
elementary symmetric functions. These are used to derive high-order, compact and parallelisable
finite difference schemes for the decay-advection-diffusion and linear damped
Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy
of the schemes is compared to that of other finite difference methods in common
use. Appendices contain useful tables of difference operators and errors and present a stability
proof for quadratic inequalities. For completeness, the appendices conclude with the
standard Thomas method for solving tri-diagonal systems.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.