Geodesic
On the global error control in nested implicit Runge--Kutta methods of Gauss type
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 3, pp. 245-259

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The automatic global error control based on a combined step size and order control presented by Kulikov and Khrustaleva in 2008 is investigated. A special attention is given to the efficiency of computation because the implicit extrapolation based on the multi-stage implicit Runge–Kutta schemes might be expensive. Especially, we discuss the technique of global error estimation and control in order to compute the numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) in the automatic mode. The theoretical results of this paper are confirmed by numerical experiments on test problems. 
@article{SJVM_2011_14_3_a1,
     author = {G. Yu. Kulikov and E. B. Kuznetsov and E. Yu. Khrustaleva},
     title = {On the global error control in nested implicit {Runge--Kutta} methods of {Gauss} type},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {245--259},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2011_14_3_a1/}
}
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G. Yu. Kulikov; E. B. Kuznetsov; E. Yu. Khrustaleva. On the global error control in nested implicit Runge--Kutta methods of Gauss type. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 3, pp. 245-259. http://geodesic.mathdoc.fr/item/SJVM_2011_14_3_a1/