Regular Runge-Kutta pairs

Time-stepping methods that guarantee to avoid spurious fixed points are said to be regular. For fixed stepsize Runge-Kutta formulas, this concept has been well studied. Here, the theory of regularity is extended to the case of embedded Runge-Kutta pairs used in variable stepsize mode with local error control. First, the limiting case of a zero error tolerance is considered. A recursive regularity test, based on the folding technique of Hairer, Iserles and Sanz-Serna (1990), is developed. It is then shown how regularity at zero tolerance carries through to the case of small tolerances. Finally, the property of regularity for all tolerances is characterized.