Two-step Runge-Kutta methods

Implicit two-step Runge-Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge-Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to y′ = ay are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge-Kutta theory where the unique two-stage method of order 4 is not semi-implicit.