Time integration of first-order ODE with higher-order information

Suppose I wish to derive a numerical integrator for the first-order ODE $$x'(t)=F(x(t)).$$ By differentiating both sides of the expression in $$t$$, I can write a second-order relation also satisfied by $$x(t)$$: $$x''(t)=\frac{d}{dt}\left(x'(t)\right)=\frac{d}{dt}F(x(t))=DF(x(t))\cdot x'(t)=DF(x(t))\cdot F(x(t)),$$ by the chain rule, where $$DF$$ denotes the Jacobian of $$F$$.

Using Taylor series, I could write $$x(t+h)=x(t)+F(x(t))\cdot h + \frac12 DF(x(t))\cdot F(x(t))\cdot h^2 + O(h^3).$$ If we omit the $$O(h^3)$$ term, we get quadratic-error explicit time integrator for our ODE.

Does this time integration strategy where we know derivatives of $$F(\cdot)$$ have a name? What are its drawbacks?

Not too surprisingly, it is referred to as a "Taylor series method." The drawback is that, in general, it may not be easy to compute the derivatives of $$F$$, particularly as you add more terms to the Taylor series to improve the accuracy.
But the method can be quite attractive if $$F$$ has a specific form, say a polynomial in $$t$$.
• In many real applications, $F$ is very complicated. Computing its derivatives is not always easy. Worse, $F$ is in many cases no differentiable -- say if a part of the device you're modeling switches on or off at specific times or based on conditions on $x$. Commented Jul 9 at 3:25
• This makes sense! I work on a lot of graphics problems where $F$ is likely to be fairly clean/differentiable --- maybe I'll play with these! Commented Jul 9 at 15:11