Strongly A-stable first stage explicit collocation methods with stepsize control for stiff and differential-algebraic equations

A variable stepsize implementation of a recently introduced one-parameter family of high order strongly A-stable Runge-Kutta collocation methods with the first internal stage of explicit type is presented. The so-called SAFERK(α,s) methods with free parameter α and s stages are well-suited for the integration of stiff and differential-algebraic systems, and they are computationally equivalent to the (s−1)-stage Radau IIA method, since they all have a similar amount of implicitness. For the same number of implicit stages, both SAFERK(α,s) and Radau IIA(s−1) methods possess algebraic order 2s−3, whereas the stage order is one unit higher for SAFERK methods. Although there are no L-stable schemes in this method family, the free parameter α can be selected in order to minimize the error coefficients or to maximize the numerical dissipation. Besides a general discussion of the method class, it is shown here how the 4-stage methods can be endowed with an embedded third order formula, and an implementation based on the perfected RADAU5 code with an adaptive stepsize controller proves to be competitive for a wide selection of test problems including electric circuit analysis, constrained mechanical systems, and time-dependent partial differential equations treated by the method of lines.