Geodesic
An efficient scheme for the implementation of implicit Runge–Kutta methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 2008-2018

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A scheme is proposed for solving nonlinear algebraic equations arising in the implementation of the implicit Runge–Kutta methods. In contrast to the available schemes, not only the starting values of the variables but also those of the derivatives are predicted. This makes it possible to reduce the number of evaluations of the function (the right-hand side) at each implicit stage without significantly reducing the accuracy of integration. 
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     author = {L. M. Skvortsov},
     title = {An efficient scheme for the implementation of implicit {Runge{\textendash}Kutta} methods},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2008--2018},
     publisher = {mathdoc},
     volume = {48},
     number = {11},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a8/}
}
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L. M. Skvortsov. An efficient scheme for the implementation of implicit Runge–Kutta methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 2008-2018. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a8/