Low order explicit Runge - Kutta formulas are quite popular for the solution of partial differential equations (PDEs) by semi-discretization, but in general-purpose codes for the solution of the initial value problem for a system of ordinary differential equations (ODES), current practice favors moderate to high order. Nevertheless, it is observed that a low order formula is more efficient at crude accuracies. Also, the stability of the formula is especially important at these accuracies, and the stability properties of explicit Runge - Kutta formulas worsen considerably as one goes to (efficient) higher order formulas. A matter of considerable importance is the availability of free interpolants for low order formulas; it is even possible to obtain interpolants that preserve qualitative properties like monotonicity and convexity l, 5. Besides the obvious value of this for plotting, these interpolants are the key to the efficient location of events G. Comparatively little attention has been devoted to low order pairs of explicit Runge - Kutta formulas. We shall mention some pairs that have been proposed, and explain why the pair we propose is either more efficient, more reliable, or has better stability. We have chosen to base our pair on a three stage, third order formula because among the minimal cost formulas, it is arguably the best with respect to stability. Also, this is the highest order for which the free shape preserving interpolants are available. Much of the solution of PDEs by semi-discretization is done with a single formula and fixed step size. We observe that the automatic control of step size with an efficient pair such as ours involves little cost per step. Not only does the control pay for itself by providing the most efficient step size, but it also avoids step sizes that lead to instability.
Bogacki, P., & Shampine, L. F. (1989). A 3(2) pair of Runge - Kutta formulas. Applied Mathematics Letters, 2(4), 321–325. https://doi.org/10.1016/0893-9659(89)90079-7