Variable Step-Size Control Based on Two-Steps for Radau IIA Methods

Two-step embedded methods of order s based on s-stage Radau IIA formulas are considered for the variable step-size integration of stiff differential equations. These embedded methods are aimed at local error control and are computed through a linear combination of the internal stages of the underlying method in the last two steps. Particular embedded methods for 2 ≤ s ≤ 7 internal stages with good stability properties and damping for the stiff components are constructed. Furthermore, a new formula for step-size change is proposed, having the advantage that it can be applied to any s-stage Radau IIA method. It is shown through numerical testing on some representative stiff problems that the RADAU5 code by Hairer and Wanner with the new strategy performs as well or even better as with the standard one, which is only feasible for an odd number of stages. Numerical experiments support the efficiency and flexibility of the new step-size change strategy.