Derivation of explicit difference schemes for ordinary differential equations with the aid of Lagrange-Burmann expansions

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.