On Low Order Embedded Pairs of Implicit Runge-Kutta Formulas

Implicit Runge-Kutta methods are used for solving stiff ODEs such as those arising in mechanical or electrical system simulation and in semidiscretisation of partial differential equation evolution problems. Embedding one Runge-Kutta formula with another is a way of obtaining an estimate of the local error (for step size control) at a modest computation cost. Our interest is with the design of embedded pairs of low order. We consider both accuracy and basic stability properties of Runge-Kutta formulas with an eye to the performance of the pair as a whole. We present some negative results showing that embedded pairs with certain combinations of stability properties cannot exist. Finally we analyze and compare 7 pairs from the literature and 6 new pairs.