# Efficient ODE steppers with query of $f$ and $\nabla f$ is efficient

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.

• You could look at Rosenbrock methods, they use one exact Jacobian per step. But I think you were looking for a method that relates to standard RK like Hermite interpolation relates to basic Lagrange interpolation? – Lutz Lehmann Dec 12 '20 at 11:14
• @LutzLehmann: Yeah, I was hoping for something like that. It appears Rosenbrock methods are all for stiff DEs, which is not my current problem, though the search term is very helpful. – user14717 Dec 12 '20 at 13:42

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 $$f$$.
• Taylor methods are explicit, and require many derivatives. If you have automatic differentiation, then if you have $f$ you have a few derivatives. Whether or not you consider them typical I guess is a matter of debate, but they are treated at length in Corless's Graduate Introduction to Numerical Methods. – user14717 Dec 14 '20 at 13:50