New answers tagged ode
3
The go-to reference on this topic is the extensive book by Hairer, Lubich, & Wanner (HLW). The kind of property you are dealing with is known as a quadratic invariant, since $\|x(t)\|^2$ is constant. These are covered in Section IV.2 of the book.
Any quadratic invariant will be preserved by a Runge-Kutta method if the method coefficients satisfy
$$
...
5
You want a numerical solution, but this might help you check your computed results.
If $a$ satisfies the ODE, you know $e^{a(t)}a'(t) = f(t)$. Integrating you get
\begin{align}
\int_0^t\, f(\tau)\, d\tau &= \int_0^t e^{a(\tau)}a'(\tau)\, d\tau \\
&= \int_{a(0)}^{a(t)} e^a da \\
&= e^{a(t)} - e^{a(0)}.
\end{align}
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