Abstract :
In [Japan JIAM 19 (2002) 227], Jackiewicz and Verner derived formulas for, and tested the implementation of two-step Runge–Kutta (TSRK) pairs. For pairs of orders 3 and 4, the error estimator accurately tracked the exact local truncation error on several nonlinear test problems. However, for pairs designed to achieve order 8, the results appeared to be only of order 6.
This deficiency was identified in [SIAM J. Numer. Anal. 34 (1997) 2087 [2]] by Hairer and Wanner who used B-series to formulate a complete set of order conditions for TSRK methods, and showed that if the order of a TSRK method is at least two greater than its stage-order, special starting values are necessary for the first step.
In forthcoming paper [Starting methods for two-step Runge–Kutta methods of stage-order 3 and order 6, J. Comput. Appl. Math.], Verner showed that such starting values have to be perturbed from their asymptotically correct values to include errors of precisely the form which the selected TSRK formula is designed to propagate from step to step. For TSRK methods of order 6, it was shown that a complementary set of Runge–Kutta methods could be utilized to obtain suitably perturbed starting values, and that each method of the set could be derived by solving appropriately modified order conditions directly. The design used there required solving an intricate polynomial equation. Here, the design is improved, and new starting methods are simpler to derive, and perhaps may lead to starting methods for TSRK methods of order 8.
