Comparison of Numerical Methods for Solving Initial Value Problems for Stiff Differential Equations

Abstract

Special classes of Initial value problem of differential equations termed as stiff differential equations occur naturally in a wide variety of applications including the studies of spring and damping systems, chemical kinetics, electrical circuits, and so on. Most realistic stiff systems do not have analytical solutions so that a numerical procedure must be used. In this paper we have discussed the phenomenon of stiffness and the general purpose procedures for the solution of stiff differential equation. Because of their applications in many branches of engineering and science, many algorithms have been proposed to solve such problems. In this study we have focused on some conventional methods namely Runge-Kutta method, Adaptive Stepsize Control for Runge-Kutta and an ODE Solver package, EPISODE. We describe the characteristics shared by these methods. We compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the traditional numerical methods such as Euler, explicit Runge-Kutta and Adams –Moulton methods step size need to be very small. This however introduces enough round-off errors to cause instability of the solution. To overcome this problem we have used two other algorithms namely Adaptive Stepsize Control for Runge-Kutta and EPISODE. The results are compared with exact one to determine the efficiency of the above mentioned method.

GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 121-132 

DOI: http://dx.doi.org/10.3329/ganit.v30i0.8509