  We deal with optimal approximation of solutions of ODEs under local Lipschitz
condition and inexact discrete information about the right-hand side functions.
We show that the randomized two-stage Runge-Kutta scheme is the optimal method
among all randomized algorithms based on standard noisy information. We perform
numerical experiments that confirm our theoretical findings. Moreover, for the
optimal algorithm we rigorously investigate properties of regions of absolute
stability.
