The Fehlberg Method

Several authors have researched on determining various sets of coefficients for higher order RK methods. Erwin Fehlberg’s NASA publications from the 1960’s (Classical Fifth, Sixth, Seventh and Eighth Order Runge Kutta Formulas With Step Size Control, NASA TR-R 287, 1968, and Low Order Runge Kutta Formulas With Step Size Control And Their Application to Heat Transfer Problems, NASA TR-R 315, 1969) present one such set of coefficients for RK methods of orders 4 and higher. The Fehlberg coefficients for RK4 are presented in the table below (ref: Hairer, Nørsett & Wanner, Solving ODEs, Vol. 1, Non-stiff problems). The generalized formulation can once again be used to solve the initial value problem.