AbstractThe 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 polynomi.....