There is no advantage to using a better ODE solver on this example because the sensitivity to errors is so high. You can measure this using the uncertainty quantification methods in the Julia ODE solver suite:
where divergence on the Lorenz attactor tends to occur by t=80 or so even with accuracy of 1e-16.
But the funny thing about chaotic problems is that the shadowing theorem holds, which is:
>Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one.
So you might as well just use Euler's method with high error, because it's backwards stable for this calculation, i.e. it gives a trajectory on the attractor, just the wrong trajectory. But since every method gives an O(1) error wrong trajectory after a short finite time, you might as well use the cheapest most error prone but convergent method.
While we are both here, if you are not aware of this paper it might give you a giggle (3500th order simulations on a supercomputer): https://arxiv.org/abs/1305.4222
They simulated Lorenz reliably up to 10000 time units; I managed 1400 units using MPFR on my NUC using my own code (500th order, took about 13 hours!).
Haha great! This is 100% a "might as well put it on Arxiv since no one reviewer would ever see this as significant" lol. It's at least very fun. I'm going to have to save this one.
https://docs.juliadiffeq.org/latest/analysis/uncertainty_qua...
where divergence on the Lorenz attactor tends to occur by t=80 or so even with accuracy of 1e-16.
But the funny thing about chaotic problems is that the shadowing theorem holds, which is:
>Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one.
So you might as well just use Euler's method with high error, because it's backwards stable for this calculation, i.e. it gives a trajectory on the attractor, just the wrong trajectory. But since every method gives an O(1) error wrong trajectory after a short finite time, you might as well use the cheapest most error prone but convergent method.