Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems

Abstract

An error analysis for a newly defined uniparametric family of stiffly accurate Runge-Kutta collocation methods when applied to initial value problems for singularly perturbed differential equations is carried out. The so-called SAFERK methods possess a first internal stage of explicit type and are based on collocation nodes. Sharp convergence results are obtained for these methods through the analysis of a sequence of higher index Differential Algebraic Equations. A numerical test with the Van der Pol oscillator reveals that the proposed error estimates are realistic whenever the stepsize h is large enough compared to the stiffness parameter e.