6th Order Runge-Kutta Pairs For Scalar Autonomous Ivp

The case of scalar autonomous Initial Value Problems (IVPs) is considered. Runge-Kutta (RK) pairs are amongst the most popular methods for solving these problems numerically. Especially RK pairs of orders 5(4) are used widely when modest accuracies required. The latter pairs need 6 stages per step when conventional methods are applied to general case IVPs. But in the scalar autonomous case we may gain an order of accuracy. Thus we may construct RK pairs of orders 6(4) for such problems at the same cost. We furnish the modified equations of condition up to seventh order for the problems of interest. We use differential evolution technique in order to derive a particular new pair of orders 6(4). Numerical tests over some well-known IVPs in the literature illustrate the efficiency of our new outcome.