High-order explicit Runge-Kutta pairs with low stage order Original Research Article

To illustrate his idea for propagating an approximate solution of an initial value problem, Runge (1895) included a pair of formulas or orders 1 and 2 (a 1,2 pair) whose difference could be used to estimate the error incurred in each step. Later work by Merson (1957) stimulated the search for p − 1, p pairs (p > 3) of Runge-Kutta formulas which would be more efficient. A scheme developed by Butcher (1963) conveniently tabulated the algebraic conditions imposed by the requirements of accuracy. Yet the direct solution of these conditions to obtain explicit Runge-Kutta pairs has remained a more challenging problem. Families of different types have been discovered through continuing research, and some of these are briefly reviewed. A recent approach (Verner, 1994) to solving the order conditions is illustrated with some new parametric families of 7,8 and 8,9 pairs.