  We study Runge-Kutta methods for rough differential equations which can be
used to calculate solutions to stochastic differential equations driven by
processes that are rougher than a Brownian motion. We use a Taylor series
representation (B-series) for both the numerical scheme and the solution of the
rough differential equation in order to determine conditions that guarantee the
desired order of the local error for the underlying Runge-Kutta method.
Subsequently, we prove the order of the global error given the local rate. In
addition, we simplify the numerical approximation by introducing a Runge-Kutta
scheme that is based on the increments of the driver of the rough differential
equation. This simplified method can be easily implemented and is computational
cheap since it is derivative-free. We provide a full characterization of this
implementable Runge-Kutta method meaning that we provide necessary and
sufficient algebraic conditions for an optimal order of convergence in case
that the driver, e.g., is a fractional Brownian motion with Hurst index
$\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting
numerical experiments verifying the theoretical rate of convergence.
