On the global error control in nested implicit Runge-Kutta methods of Gauss type

Numerical Analysis and Applications, 14(3) (2011), 245-259
http://dx.doi.org/10.1134/S1995423911030025

Automatic global error control based on a combined control of step size and order presented by Kulikov and Khrustaleva in 2008 is investigated. Special attention is given to the efficiency of computation, because the implicit extrapolation based on multistage implicit Runge-Kutta schemes may be expensive. Specifically, we discuss a technique of global error estimation and control in order to compute a numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) automatically. The theoretical results of this paper are confirmed by numerical experiments on test problems.