Reformulate ARK

[Sbozzolo]

Work in progress

Closes #299

[charleskawczynski]

I think this may have been an incorrect merge resolution: save_func is not documented anywhere (it seems to be internal), so we shouldn't use it. Also, I don't like the idea of using DiffEqBase.KeywordArgSilent.

[Sbozzolo]

@dennisYatunin

[dennisYatunin]

The convergence tests were all failing with the callback solution on main. I agree that this should be done in terms of callbacks, though. Gabriele, maybe you could update this while ensuring that the convergence tests still pass?

[charleskawczynski]

They were failing before this change. They've been periodically breaking on main for months. So, it's not clear to me that this needs to be reverted.

[charleskawczynski]

In fact, I don't believe that these changes should change the results, so I'd be surprised if this was responsible for the docs breaking.

[dennisYatunin]

I fixed all convergence tests in this rebase, and I was not able to get them running without reverting this. Granted, I didn't spend too much time debugging the callback version, so it might have been a simple issue.

[dennisYatunin]

It would also be good to add @test or @assert calls to the convergence tests to avoid having them silently break for months.

[charleskawczynski]

The problem is that it's flakey, it passes locally, or on push, and then fails on the main branch.

Another issue is that virtually nothing is printed when it errors, it's not clear which example is breaking, and what any of the types are. I think a better solution would be to run these examples on buildkite, and have a dependent job make a summary.

I don't want to add the use of internals back into this repo, so please fix the issue without save_func /KeywordArgSilent, or fix the underlying issue, or leave it broken, as it has been (and very well may still be in this repo as the issue is not easily reproducible).