Geodesic
Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems
Matematičeskoe modelirovanie, Tome 23 (2011) no. 1, pp. 87-99

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Two schemes of the third and fourth order are presented on the basis of the schemes including the right parts derivatives. The iterative procedure is suggested for the realization of these schemes combined with using of the complex-valued arithmetic. Sufficiency of only two Newtonian iterations is shown for the one time step calculation. The procedure for local error estimation based on results of two sequential iterations has been proposed and approved.
Keywords: stiff differential equations systems, implicit schemes with derivatives, complex-valued realization. 
@article{MM_2011_23_1_a7,
     author = {Yu. A. Sigunov and I. R. Didenko},
     title = {Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the {ODE} systems},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {87--99},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/}
}
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Yu. A. Sigunov; I. R. Didenko. Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems. Matematičeskoe modelirovanie, Tome 23 (2011) no. 1, pp. 87-99. http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/