Geodesic
Derivation of explicit difference schemes for ordinary differential
Numerical methods and programming, Tome 11 (2010) no. 2, pp. 198-209.

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Some explicit multistage Runge-Kutta type methods for solving ordinary differential equations (ODEs) are derived with the aid of the expansion of grid functions in the Lagrange-Burmann series. The formulas are given for the first four coefficients of the Lagrange-Burmann expansion. New explicit first- and second-order methods are derived and applied to the numerical integration of the Cauchy problem for a moderately stiff ODE system. It turns out that the $L_2$-norm of the error in the solution obtained by the new numerical second-order method is 50 times smaller than that of the classical second-order Runge-Kutta method.
Keywords: ordinary differential equations; Lagrange-Burmann expansion; Runge-Kutta methods; stiff systems. 
@article{VMP_2010_11_2_a6,
     author = {E. V. Vorozhtsov},
     title = {Derivation of explicit difference schemes for ordinary differential},
     journal = {Numerical methods and programming},
     pages = {198--209},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2010_11_2_a6/}
}
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E. V. Vorozhtsov. Derivation of explicit difference schemes for ordinary differential. Numerical methods and programming, Tome 11 (2010) no. 2, pp. 198-209. http://geodesic.mathdoc.fr/item/VMP_2010_11_2_a6/