Geodesic
Singly implicit diagonally extended Runge–Kutta methods of fourth order
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 5, pp. 755-765

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Singly implicit diagonally extended Runge–Kutta methods make it possible to combine the merits of diagonally implicit methods (namely, the simplicity of implementation) and fully implicit ones (high stage order). Due to this combination, they can be very efficient at solving stiff and differential-algebraic problems. In this paper, fourth-order methods with an explicit first stage are examined. The methods have the third or fourth stage order. Consideration is given to an efficient implementation of these methods. The results of tests in which the proposed methods were compared with the fifth-order RADAU IIA method are presented. 
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     author = {L. M. Skvortsov},
     title = {Singly implicit diagonally extended {Runge{\textendash}Kutta} methods of fourth order},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {755--765},
     publisher = {mathdoc},
     volume = {54},
     number = {5},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_5_a4/}
}
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L. M. Skvortsov. Singly implicit diagonally extended Runge–Kutta methods of fourth order. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 5, pp. 755-765. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_5_a4/