I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel transport of a tangent vector in closed form (For example, on the sphere $\mathbb{S}^{2} \subset \mathbb{R}^{3}$, the parallel transport of a tangent vector along a curve (not necessarily geodesic) can be obtain in closed form.) We can only obtain an approximation of this transport through a numerical scheme.

For example, the Schild's Ladder (see Schild's Ladder Parallel Transport Procedure for an arbitrary connexion) is a first-order numerical scheme which allows to approximate the parallel transport of a tangent vector along a curve.

My questions are the following (I apologize if these questions are naive) :

- Is there a reference, a proof of how fast this approximation converges to the exact parallel transport ?
- Do you know if there exists a second-order approximation of the parallel transport ?