Construction of explicit Runge--Kutta pairs with stiffness detection

Explicit Runge-Kutta schemes are the methods of choice for solving non-stiff systems of ordinary differential equations at low to medium tolerances. The construction of optimal formulas has been the subject of much research. In this article it will be shown how to construct some low order formula pairs using tools from computer algebra. Our focus will be on methods that are equipped with local error detection (for adaptivity in the step size) and with the ability to detect stiffness. It will be demonstrated how criteria governing `optimal' tuning of free parameters and matching of the embedded method can be accomplished by forming a constrained optimization problem. In contrast to standard numerical optimization processes our approach finds an exact (infinite precision) global minimum. Quantitative measures will be given comparing our new methods with some established formula pairs.