Research

Dr. Noorhelyna Razali
Title:
The symmetrization of Lobatto IIIA methods in solving stiff ordinary differential problems in engineering
Summary:
Among the models using ordinary differential equations are various physical problems such as the Kepler problem, the simple pendulum, electrical circuits, the study of vibrations, biomechanical systems, weather forecasting and the chemical kinetics problems. In the real-world situation, certain problems can be ‘difficult’ to solve analytically, hence numerical methods are required to provide approximate solutions. We study two-step symmetrization of Runge–Kutta methods for stiff ordinary differential equations. The process is a generalization of the smoothing formula introduced by Gragg in 1965. It can be regarded as being generated by a related Runge–Kutta method constructed so as to preserve the asymptotic error expansion in even powers of the stepsize. When the method is applied with an accelerating technique such as extrapolation, the order can then be increased by two at
each level of extrapolation. The method is also L-stable and hence provides damping for stiff problems. In this research, we will extend our investigation to two-step symmetrization and consider the application in various problems for example in Van der Pol, HIRES, OREGO, and BRUS problems. We will apply the symmetrization process in both constant and variable stepsize settings and test the efficiency and accuracy of the methods for linear and nonlinear problems. We also extend the application to the higher order method.