Fifth-order improved Runge-Kutta method with reduced number of function evaluations

The Improved Runge-Kutta (IRK) methods are two step in nature and require lower number of stages per step compared to the classical Runge-Kutta methods. Therefore, the IRK methods will have the lower number of function evaluations per step. Here, the fifth-order Improved Runge-Kutta method (IRK5) with only five stages is derived. The order conditions of the method are obtained up to order six and the coefficients of the fifth order method are determined by minimizing the error norm of the sixth order method. The stability region of the method is presented and numerical examples are given to illustrate the computational efficiency and accuracy of IRK5 compared to RK5.