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

• Published in: Journal of computational and applied mathematics 2014-03, Vol.259, p.138-152
• Main Authors: González-Pinto, S., Hernández-Abreu, D., Simeon, B.
• Format: Article
• Language: English
• Subjects: Adaptive integration; Differential–algebraic equations; RADAU5; Runge–Kutta collocation methods; SAFERK methods; Stiff systems
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2013.04.011

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.