Unequally spaced knot techniques for the numerical solution of partial differential equations.

Cubic spline approximations to time dependent partial differential equations, having both constant and variable coefficients, are developed in which the knot points may be chosen to be unequally spaced. Four methods are presented for obtaining 'optimal' knot positions, these being chosen so at to produce an increase in accuracy compared with methods based on equally spaced knots. Three of the procedures described produce knot partitions which are fixed throughout time. The fourth procedure yields differently placed 'optimal' knots on each time line, thus enabling us to better approximate the varying time nature characteristic of many partial differential equations. Truncation errors and stability criteria are derived and full numerical implementation procedures are given. Five case studies are presented to enable comparisons to be drawn between the knot placement methods and results found using equally spaced knots. Possible extensions of the work of this thesis are given.