Geodesic
$L(\alpha)$-stable variable order implicit second derivative Runge Kutta methods
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 17 (2014) no. 4, pp. 373-387

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This paper considers the extension of the popular Runge Kutta methods (RKMs) to second derivative Runge Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation techniques. The last stage of the input approximation is identical to the output method. The SDRKMs are $L(\alpha)$-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM). 
@article{SJVM_2014_17_4_a5,
     author = {R. I. Okuonghae and M. N. O. Ikhile},
     title = {$L(\alpha)$-stable variable order implicit second derivative {Runge} {Kutta} methods},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {373--387},
     publisher = {mathdoc},
     volume = {17},
     number = {4},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2014_17_4_a5/}
}
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R. I. Okuonghae; M. N. O. Ikhile. $L(\alpha)$-stable variable order implicit second derivative Runge Kutta methods. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 17 (2014) no. 4, pp. 373-387. http://geodesic.mathdoc.fr/item/SJVM_2014_17_4_a5/