Fourth- and Fifth-Order, Scaled Rungs–Kutta Algorithms for Treating Dense Output

Abstract

New Runge–Kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step, as well as at the end of each step. The new scaled Runge–Kutta methods are designed to be used with existing Runge–Kutta formulas, using the derivative evaluations of these defining algorithms as the core of the new system. Thus, for only a slight increase in computing time, the solution may be generated within the integration step, improving the efficiency of the Runge–Kutta algorithms, since the step length need no longer be reduced to coincide with the desired output point. Scaled Runge–Kutta algorithms are presented for orders 4 and 5, along with accuracy comparisons between the defining algorithms and their scaled versions for a test problem.