Numerical integrator for $a'(t)=e^{-a(t)}f(t)$

Question

Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ satisfies the ODE: $$\frac{da(t)}{dt}=e^{-a(t)}f(t).$$ I'm willing to compute (in closed-form) any derivatives $f^{(k)}(t)$ if that helps.

Of course I can apply the usual explicit methods (forward Euler, RK3), but I'm hoping to find an integrator that's reversible in time. And, ideally, one that exploits special structure in the ODE above, e.g. by somehow integrating the exponential, to achieve higher accuracy on some model problems.

Any suggestions?

Answer 1

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}