The primary aim of this thesis is to calculate the numerical solution of a given stiff system of ordinary differential equations. We deal with the implementation of the implicit Runge-Kutta methods, in particular for Radau IIA order 5 which is now a competitive method for solving stiff initial value problems. New software based on Radau IIA, called IRKMR5 written in MATLAB has been developed for fixed order (order 5) with variable stepsizes, which is quite efficient when it is used to solve stiff problems. The code is organized in a modular form so that it facilitates both the understanding of it and its modification whenever needed. The new software is not only more functional than its Fortran 77 Radau IIA counterpart but also more robust and better documented. When implicit methods are used to solve nonlinear problems it is necessary to solve systems of nonlinear algebraic equations. New investigations for a modified Newton iteration are undertaken. This new strategy manages the iterative solutions of nonlinear equations in the ODEs solver. It also involves when to re-evaluate the Jacobian and the iteration matrix. The strategy also significantly reduces the number of function evaluations and linear solves. We subsequently consider the mathematical analysis of the nonlinear algebraic equations that arise from using s-stage fully implicit Runge-Kutta methods. Results for uniqueness of solutions and an error bound was established. The termination criterion in the iterative solution of the nonlinear equations is also studied as well as two types of termination criterion (displacement and the residual test). The residual test has been compared with the displacement test on some test examples and the results are tabulated.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:488322 |
| Date | January 1999 |
| Creators | Al-Harbi, Saleh M. |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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