Starting methods for two-step Runge–Kutta methods of stage-order 3 and order 6

Jackiewicz and Tracogna [SIAM J. Numer. Anal. 32 (1995) 1390–1427] proposed a general formulation of two step Runge–Kutta (TSRK) methods. Using formulas for two-step pairs of TSRK methods constructed in [Japan JIAM 19 (2002) 227–248], Jackiewicz and Verner obtain results for order 8 pairs that fail to show this designated order. Hairer and Wanner [SIAM J. Numer. Anal. 34 (1997) 2087–2089] identify the problem by using B-series to formulate a complete set of order conditions for TSRK methods, and emphasize that special starting methods are necessary for the first step of implementation. They observe that for methods with stage order at least p - 1 , and design order p , starting methods of order at least p are sufficient. In this paper, the more general challenge to provide correct starting values for methods of low stage-order is met by showing how perturbed starting values should be selected for methods of order 6 and stage-order 3. The approach is sufficiently general that it may (and later will) be provided for such methods of higher orders. Evidence of the accompanying improvement in the implementation of TSRK methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage-order.