  In Spivak's work, dynamical systems are described in terms of their inputs
and outputs in a pictorial way using an operad of wiring diagrams. Each
dynamical system is a box with certain inputs and outputs, and multiple
dynamical systems are linked together using wiring diagrams, which describe how
the outputs of one dynamical system to the inputs of another. By describing
dynamical systems in this way, we can decompose a large dynamical system as a
collection of smaller, simpler dynamical systems linked together. Of course,
this decomposition is only useful if we can work with these smaller, simpler
dynamical systems instead of the larger one. In his paper, Spivak shows that we
can perform Euler's method on these smaller systems and still get the same
results as working on the larger one. In this paper, we extend his results to
prove that we can do something similar with the Runge-Kutta method. However, we
need to modify the framework used in Spivak's paper to account for the fact
that the Runge-Kutta method requires multiple steps, unlike Euler's method. To
better describe these systems, we define wiring diagrams as objects of a
double-category and dynamical systems in terms of double functors, giving a
categorical description of this approximation method.
