# 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? 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. Dec 12 '20 at 13:42

## 2 Answers

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.

• 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. Dec 14 '20 at 13:50
• @user14717 But Taylor methods are not widely used, for stability reasons and also because they can not be used if the right hand side is not a function with many bounded derivatives (which is indeed not the case for many, maybe most, practical problems). Dec 14 '20 at 18:01

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.