A Special Class of Multistep Runge-Kutta Methods with Extended Real Stability Interval

A special class of fc-step Runge-Kutta methods is investigated which is generated by (non-linear) Chebyshev iteration (Richardson iteration) of an implicit linear multistep method. By terminating the iteration process after (say) m iterations, a family of k-step, m-stage Runge-Kutta methods is obtained for which the real stability interval can be derived for general values of k and m by a special application of the boundary locus method. The real stability boundary is maximized by choosing suitable values for the coefficients in the generating fc-step method. The investigation is mainly restricted to second-order methods. Examples are given for k = 1,2,3 and 4, and a few numerical experiments with non-linear parabolic initial-boundary value problem