# Efficiency of Extrapolated Runge-Kutta Methods in Solving Linear and Nonlinear Problems

## Kecekapan Extrapolasi Kaedah Runge-Kutta dalam Menyelesaikan Masalah Linear dan Tak Linear

## Keywords:

Runge-Kutta methods, symmetric methods, smoothing, rational and polynomial extrapolation## Abstract

Extrapolation involves taking a certain linear combination of the numerical solutions of a base method applied with different stepsizes to obtain greater accuracy. This linear combination is done to eliminate the leading error term. The technique of extrapolation (passive and active) in accelerating convergence has been used successfully in numerical solution of ordinary differential equations. In this study, symmetric Runge-Kutta methods for solving linear and nonlinear stiff problem are considered. Symmetric methods admit asymptotic error expansion in even powers of the stepsize and are therefore of special interest because successive extrapolations can increase the order by two at time. Two ways of applying extrapolation are considered such as the active and the passive. It is interesting to know which modes of applying extrapolation are the most efficient when applied with symmetric methods. Results of numerical experiments are given which show the efficiency of the rational and polynomial extrapolated Implicit Midpoint Rule (IMR) and the Implicit Trapezoidal Rule (ITR) in solving Chemistry and Chemical Reaction problems. Numerical results show that in both types of extrapolation, the passive mode is considered to be the most efficient. The results also show that extrapolation with smoothing gives better results than without smoothing.

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*Journal of Science and Mathematics Letters*,

*1*, 1–8. Retrieved from https://ejournal.upsi.edu.my/index.php/JSML/article/view/4300