Sains Malaysiana 47(11)(2018): 2927–2932 http://dx.doi.org/10.17576/jsm-2018-4711-36 Comparison of One-Step and Two-Step Symmetrization in the Variable Stepsize Setting (Perbandingan Satu dan Dua Langkah Pensimetrian dalam Persekitaran Saiz Langkah Berubah-Ubah) N. RAZALI*, Z.M. NOPIAH & H. OTHMAN ABSTRACT In this paper, we study the effects of symmetrization by the implicit midpoint rule (IMR) and the implicit trapezoidal rule (ITR) on the numerical solution of ordinary differential equations. We extend the study of the well-known formula of Gragg to a two-step symmetrizer and compare the effciency of their use with the IMR and ITR. We present the experimental results on nonlinear problem using variable stepsize setting and the results show greater effciency of the two-step symmetrizers over the one-step symmetrizers of IMR and ITR. Keywords: Implicit midpoint rule; implicit trapezoidal rule; symmetrizers ABSTRAK Dalam kertas ini, kami mengkaji kesan pensimetrian kaedah titik tengah tersirat (IMR) dan kaedah trapezium tersirat (ITR) ke atas penyelesaian berangka persamaan pembezaan biasa. Kami melanjutkan kajian terkenal oleh Gragg kepada pensimetri dua langkah dan membandingkan kecekapan penggunaannya dengan IMR dan ITR. Keputusan uji kaji pada masalah tidak linear menggunakan saiz langkah yang berubah-ubah menunjukkan bahawa pensimetri dua langkah adalah lebih cekap berbanding pensimetri satu langkah. Kata kunci: Kaedah titik tengah tersirat; kaedah trapezium tersirat; pensimetri INTRODUCTION The study of numerical methods for initial value problems including stiff problems has introduced several new ideas on stability and error propagation. Stiff ordinary differential equation systems arise in many different application areas where the components of the solution have widely different rates of change. These problems involving rapidly decaying transient components as well as steady-state ones occur naturally in many different situations including the damped spring system, control systems and chemical kinetics. Some studies on stiff problems have been reported in Auzinger and Macsek (1990), Bjurel et al. (1970), Burrage (1978), Enright et al. (1975), Liniger and Willoughby (1970) and Mazzia et al. (2012). We are interested in solving ordinary differential equations, especially stiff problems. Symmetric Runge- Kutta methods are considered because their numerical solution possesses an asymptotic error expansion in even powers of the stepsize h. When applied with Richardson extrapolation the order can potentially increase by two at each level of extrapolation. Gragg (1965) frst proved the existence of an asymptotic error expansion for the explicit midpoint rule which laid the foundation for the application of Richardson extrapolation. He also introduced the concept of smoothing to suppress the effects of the parasitic oscillatory component in the numerical solution. Since then, the concept of extrapolation and smoothing has been applied in nonstiff and stiff problems by many researchers. Following the idea of Gragg, Chan (1989) generalized the smoothing technique for arbitrary symmetric Runge-Kutta methods. He called the process symmetrization and showed how it can be achieved by an L-stable method known as the symmetrizer that is constructed so as to preserve the asymptotic error expansion in even powers of stepsize and to provide the necessary damping for stiff problems. In 2012, Gorgey extended Chan’s theoretical study of extrapolation and symmetrization by means of practical implementation and experimental study. She investigated the two modes of symmetrization, that are, active and passive symmetrizations of one-step symmetrizers for Gauss and Lobatto IIIA methods of order 4 and 6 in the constant and variable stepsize settings. She analyzed the most effcient way of implementing symmetrization with and without extrapolation on order-4 and order-6 methods, providing evidence that the one-step symmetrizers can restore the classical order especially the order-4 Gauss and Lobatto IIIA methods (Chan & Gorgey 2013, 2011). Symmetrization can be applied in different ways. In a constant stepsize setting, the possibilities are: Active Symmetrization: Each time a symmetrized value is computed it is then used to propagate the numerical solution. Symmetrization can be performed at every step, every two steps or every three steps.