Ishak, Fuziyah
(2009)
Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations.
PhD thesis, Universiti Putra Malaysia.
Abstract / Synopsis
This thesis describes the development of sequential and parallel methods for solving
delay differential equations. A new sequential code for the numerical solution of delay
differential equations is considered. The variable order variable stepsize formulae
based on the AdamsBashforthMoulton methods are represented in divided difference
form. Derivative discontinuities are detected by local error estimate at the grid points.
Large magnitude of the local error estimate indicates the presence of derivative discontinuity.
Stepsize is then reduced and eventually the discontinuity point is included in
the grid. The formulae representation proves to be efficient when compared with the
existing method in modified divided difference form.
We also consider the development of twopoint block methods on sequential and parallel
computers. Formulae for three twopoint block methods for solving delay differential
equations are derived. The implicit block methods are implemented using variable
stepsize variable order technique. The formulae for twopoint diagonally and triangu larly implicit block methods using predictorcorrector application are represented in
divided difference form. Meanwhile, the predictorcorrector formulae for twopoint
fully implicit block method are calculated beforehand and stored at the beginning of
the code. All of the block methods rely on the local error estimates to detect derivative
discontinuities. In all of the developed methods, regions of absolute stability are
presented and compared. Comparison among the developed methods indicates that all
of the methods achieve the desired accuracy. Block methods are efficient when compared
with the sequential nonblock method as the total steps taken can be reduced.
The new block methods are then used for the parallel implementation in solving large
system of delay differential equations. The parallel programs using Message Passing
Interface are run on Sun Fire V1280 using two processors. Numerical results indicate
that parallel implementation increases the performance of the block methods.
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