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In this paper, we shall study the convergence of Taylor approximations forthe backward Loewner differential equation (driven by Brownian motion) near theorigin. More concretely, whenever the initial condition of the backward Loewnerequation (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 theTaylor expansion become larger than "lower degree" terms for small$\varepsilon$. In this sense, the time horizon on which approximations areaccurate scales like $\varepsilon^{2}$. This scaling comes naturally from theLoewner equation when growing vector field derivatives are balanced againstdecaying iterated integrals of the Brownian motion. As well as being oftheoretical interest, this scaling may be used as a guiding principle fordeveloping adaptive step size strategies which perform efficiently near theorigin. In addition, this result highlights the limitations of using stochasticTaylor methods (such as the Euler-Maruyama and Milstein methods) forapproximating $SLE_{\kappa}$ traces. Due to the analytically tractable vectorfields of the Loewner equation, we will show Ninomiya-Victoir (or Strang)splitting is particularly well suited for SLE simulation. As the singularity atthe origin can lead to large numerical errors, we shall employ the adaptivestep size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces usingthis splitting. We believe that the Ninomiya-Victoir scheme is the first highorder numerical method that has been successfully applied to $SLE_{\kappa}$traces.

An asymptotic radius of convergence for the Loewner equation and simulation of $SLE_k$ traces via splitting