Abstract:
A new type of general linear method for the numerical solution of stiff differential
equations has been discovered recently. These methods are characterised by a
property known as "inherent Runge-Kutta stability". This property implies that
the stability matrix has only a single non-zero eigenvalue and that this eigenvalue
is a rational approximation to the exponential function. This rational approximation
is just like the stability function of a Runge-Kutta method. The theoretical
properties of the new methods, such as stability and order, are surveyed. Also
constructing practical general linear methods of this type for stiff problems is
discussed.
The emphasis of the thesis is on implementation and numerical experiments.
We have investigated several implementation questions such as starting methods,
prediction of the stage values, an efficient iteration scheme, truncation error estimation
and stepsize control. For each of these questions, numerical tests have
been carried out.
To investigate a fixed order general linear method, a starting method is needed
to provide the first incoming approximations. It is to be expected that a good
prediction of the initial iteration in the Newton method calculations will reduce
the number of iterations required. Since the LU factorizations are expensive, we
have designed an efficient iteration scheme to reduce the computational cost. A
reliable error estimator gives an optimal stepsize sequence. These implementation
issues are addressed in detail.
