Are there numerical methods for ODEs that are capable of producing regions (linearly bounded or otherwise) that definitely contain a portion of a specified integral curve?
I'm wondering if there are numerical methods for solving ODEs that are roughly analogous to interval arithmetic for real-valued functions.
Suppose we have two unknowns $y(x)$ and $x$ where $y' = F(x, y)$. Suppose that $(x_0, y_0)$ is our initial value and suppose that $-1 \le x \le 1$ is the range of values of $x$ that we're interested in. In other words, $(x_0, y_0)$ is how we pick out a particular integral curve and $S \stackrel{\text{def}}{=} \left\{ (x,y) : x, y \in \mathbb{R} \land -1 \le x \le 1 \right\}$ is the region of the $xy$ plane that we want to pick a subset of.
Are there any methods that produce a subset of $S$ that the integral curve passing through $(x_0, y_0)$ must reside in that is smaller than $S$ itself, preferably one where the area covered by the estimate region can be made arbitrarily small by picking a step size or something similar?