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Assume we have an IVP $y'(t) = f(t,y)$, and that $\partial_t f$ and $\nabla f$ are cheap to compute.
Assume further that more derivatives are not cheap to compute, or inaccessible for some reason, perhaps API related.
What are the best ODE steppers for this case?
Taylor series methods are available with $O(h^3)$ error, but presumably assuming more smoothness we can do better.
None of the typical ODE integrators ever need more than first derivatives of $f$. In fact, all explicit methods, including Runge-Kutta methods, only require knowledge of $f$ itself. Implicit methods require you to solve a nonlinear system in which $f$ appears, and that is done using (variations of) Newton's method which will only ever require first derivatives of $f$.
In other words, you will not ever need anything more than what you already have. That leaves you with a large number of possible methods. To choose which one you have, you need to think about the properties of the ODE (e.g., stiff/nonstiff, stable/non-stable) and whether that warrants the use of explicit or implicit integrators, which order you need, etc.
Methods of this sort are commonly called "Obrechkoff methods", or "Enright methods". Some examples are designed by Enright, another by Wu, and are reviewed by Huang and Innanen.
