Parallel Block Methods for Solving Ordinary Differential Equations - Abdul Majid, Zanariah (2004)
In this thesis, new and efficient codes are developed for solving Initial Value Problems (IVPs) of first and higher order Ordinary Differential Equations (ODEs) using variable step size. The new codes are based on the implicit multistep block methods formulae. Subsequently, a more structured and efficient algorithm comprising the block methods was constructed for solving systems of first order ODEs using variable step size and order. The new codes were then used for the parallel implementation in solving large systems of first and higher order ODEs. The sequential programs of these methods were executed on DYNIXlptx operating system. The parallel programs were run on a Sequent Symmetry SE30 parallel computer.The Cq stability in the multistep method was introduced and the focused was on the error propagation from a more practical angle. The numerical results showed that the sequential implementation of the new codes could reduce the total number of steps and execution times even when solving small systems of first and higher order ODEs compared with the 1-point method and the existing 2PBVSO code in Omar (1 999). The parallel implementation of the codes was found to be most appropriate in solving large systems of first and higher order ODEs. It was also discovered that the maximum speed up of the parallel methods improved as the dimension of the ODEs systems increased. In conclusion, the new codes developed in this thesis are suitable for solving systems of first and higher order ODEs