Order-2 extrapolated implicit runge-kutta methods with smoothing for solving ordinary differential equations

The aim of this research is to study the efficiency of the order-2 extrapolated implicit Runge-Kutta methods. The order-2 methods being considered are the implicit midpoint (IMR) and implicit trapezoidal (ITR) rules. These methods are applied with the polynomial and rational extrapolations actively and passively with smoothing to improve the accuracy. In order to reduce the round-off errors, a technique known as compensated summation with simplified Newton is implemented in all the numerical codes. The results are given based on numerical experiments that are carried out using MATLAB software. The findings showed that passive polynomial extrapolation by the IMR and ITR are more efficient than with active and passive rational extrapolation. The smoothing technique with extrapolation by the IMR and ITR gives better behavior than without the smoothing technique. It is therefore concluded that in solving chemistry, linear and nonlinear chemical and logistic curve problems, passive polynomial extrapolation with smoothing (PPXS) by the IMR gives better efficiency than the passive and active polynomial extrapolation by the 2-stage Radau IIA method (R2PX) and (R2AX). However, for higher dimensional nonlinear problems, R2AX and R2PX can be as efficient as the PPXS. The implication of this study is that, PPXS that has a cheaper implementation cost can be a very efficient method in solving linear and nonlinear stiff problems when compared with other higher order methods. Therefore, it is recommended to apply IMR with smoothing and extrapolation in the future research for comparison involving lower order methods.