Design and construction of software for general linear methods : a thesis submitted in fulfilment of the requirements of Doctor of Philosophy in Mathematics, New Zealand Institute for Advanced Studies (NZIAS), Massey University, Albany Campus, Auckland, New Zealand
The ultimate goal in the study of numerical methods for ODEs is the construction of such methods
which can be used to develop efficient and robust solvers. The theoretical study and investigation
of stability and accuracy for a particular class of methods is a first step towards the development of
This thesis is concerned with the use of general linear methods (GLMs) for this purpose. Whereas
existing solvers use traditional methods, GLMs are more complex due to their complicated order
conditions. A natural approach to achieve practical GLMs, is to first consider the advantages and
disadvantages of traditional methods and then compare these with a particular class of GLMs. In
this thesis, GLMs with IRKS– and F–properties are considered within the type 1 DIMSIMs class. The
freedom of choice of free parameters in IRKS methods is used here to test the sensitivity and capability
of the methods.
A complete ODE software package uses many numerical techniques in addition to the methods
considered. These include error estimation, interpolation for continuous output, etc.. Existing ODE
software is a combination of these techniques and much work has been done in the past to improve the
capability of these traditional methods. The approach has been largely heuristic and empirical. These
are developed by fitting all these techniques into one algorithm to produce efficient ODE software.
The design of the algorithm is the main interest in the thesis. An efficient solver will be in (h, p)-
refinement mode. This design includes many decisions in the whole algorithm. These include selection
of stepsize and order for the next step, rejection criteria, and selection of stepsize and order in case of
rejection. To design such a robust algorithm, the Lagrange optimisation approach is used here. This
approach for the selection of stepsize and order avoids the use of several heuristic choices and gives a
new direction for developing reliable ODE software. Experiments with this design have been carried
out on non–stiff, mildly–stiff and some discontinuous problems and are reported in this thesis.