  In this paper, we shall study the convergence of Taylor approximations for
the backward Loewner differential equation (driven by Brownian motion) near the
origin. More concretely, whenever the initial condition of the backward Loewner
equation (which lies in the upper half plane) is small and has the form $Z_{0}
= \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$
error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$.
Statements of this theorem will be given using both rough path and
$L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of
$\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the
Taylor expansion become larger than "lower degree" terms for small
$\varepsilon$. In this sense, the time horizon on which approximations are
accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the
Loewner equation when growing vector field derivatives are balanced against
decaying iterated integrals of the Brownian motion. As well as being of
theoretical interest, this scaling may be used as a guiding principle for
developing adaptive step size strategies which perform efficiently near the
origin. In addition, this result highlights the limitations of using stochastic
Taylor methods (such as the Euler-Maruyama and Milstein methods) for
approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector
fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang)
splitting is particularly well suited for SLE simulation. As the singularity at
the origin can lead to large numerical errors, we shall employ the adaptive
step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using
this splitting. We believe that the Ninomiya-Victoir scheme is the first high
order numerical method that has been successfully applied to $SLE_{\kappa}$
traces.
