Comparison of Some Numerical Methods for Stiff Systems of Ordinary Differential Equation

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The phenomenon of stiffness arises whenever a system of differential equations has two or more solution components with rapidly varying time-scales of the independent variable Practical problems and applications in fields such as Mechanics, Control theory, Chemical and Nuclear kinetics , Electronics, and Polution models often lead to stiff systems of differential equations which must be solved. Significant difficulties, however, often arise when standard numerical methods are applied to approximate the desired solutions of these important class of problems. It is therefore desirable to investigate some of the practical and efficient numerical methods for solving such problems. Although a number of methods have been developed and many more basic formulas suggested for stiff differential equations, advice or guidance to assist a user or practitioner choose a suitable method or his/her problem has not been given the desired attention until recently. Thus a thorough investigation of some of the available routines to ascertain their relative efficiency and performance on different classes of stiff problems is also a step in the right direction. Some of the methods for stiff systems we considered in this 5tudy include the BDF—based methods, Generalized Implicit and Semi- implicit fl-methods, Second Derivative Formula--based methods (also called Exponential Fitting methods), and the Modified Semi- implicit Extrapolation methods. At the preliminary investigation stage, we found out that the classical explicit methods both of the Runge-Kutta and Linear Multistep type are extremely inefficient on stiff problems. Their application may, however, find a secure place in the solution of non-stiff problems. We also found out that generally the Implicit Linear Multistep methods of the Adams-Moulton type perform very poorly on stiff problems. We do not therefore recommend this class of method for stiff problem. Our intermediate investigation showed that the BDF-based method, the Second Derivative-based method, the Implicit RK- method of butcher, the Generalised Implicit and Semi – implicit RK-method, the Semi – implicit Extraplation method of Bader and Deuflhard, and the Rosenbrock method of Kape and Rentrop, all perform quite satisfactorily with varying degree of success on various classes of stiff problems. The method of Bader/Deuflhard and Kape/Rentrop proved to be efficient than the more conventional method. Our final investigation leads us to recommend the Bader-Deuflhard Extrapolation method for most stiff problems, because of its greater success in solving various classes of such problems. For less stringent accuracy requirement and mildly stiff problem, however, we do recommend the use of the Kaps-Rentrop because of its ease of implementation and because it compares favourably with the more exhortic conventional code available. For stiff chemical problems however, the BDF-based methods remain very efficient. We therefore recommend the package GEAR, REV3,and the second Derivative – based methods such as SDBASIC of Enright for such problems.
A thesis submitted to the Board of Postgraduate Studies, Kwame Nkrumah University of Science and Technology, Kumasi, in partial fulfilment of the requirement for the award of the Degree of Master of Science in Mathematics, 1998