  Classical and new numerical schemes are generated using evolutionary
computing. Differential Evolution is used to find the coefficients of finite
difference approximations of function derivatives, and of single and multi-step
integration methods. The coefficients are reverse engineered based on samples
from a target function and its derivative used for training. The Runge-Kutta
schemes are trained using the order condition equations. An appealing feature
of the evolutionary method is the low number of model parameters. The
population size, termination criterion and number of training points are
determined in a sensitivity analysis. Computational results show good agreement
between evolved and analytical coefficients. In particular, a new fifth-order
Runge-Kutta scheme is computed which adheres to the order conditions with a sum
of absolute errors of order 10^-14. Execution of the evolved schemes proved the
intended orders of accuracy. The outcome of this study is valuable for future
developments in the design of complex numerical methods that are out of reach
by conventional means.
