[Proof of Concept] Make ForwardDiff work with implicit time integration

[efaulhaber]

A beautiful solution for #2369.

I think it's pretty clear that I have absolutely no idea what I'm doing.
I have no idea where the numbers 1 and 11 come from (are these the chunk sizes?), but it works like that for this specific elixir.

We would need to know all dual types that the rhs! will be called with, and then we can construct a cache (or in this case a whole semidiscretization) for each of them.
No idea how to implement this in a clean way, but at least it demonstrates that this can be done and especially that it can be done in a minimally invasive way without changing all the data structures.

[github-actions]

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[efaulhaber]

CC @vchuravy

[codecov]

Codecov Report

Attention: Patch coverage is 28.57143% with 5 lines in your changes missing coverage. Please review.

Project coverage is 96.87%. Comparing base (afdb486) to head (0fa2ecb).

Files with missing lines	Patch %	Lines
...amples/tree_1d_dgsem/elixir_advection_diffusion.jl	0.00%	3 Missing ⚠️
...emidiscretization/semidiscretization_hyperbolic.jl	50.00%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2371      +/-   ##
==========================================
- Coverage   96.93%   96.87%   -0.06%     
==========================================
  Files         501      501              
  Lines       41589    41596       +7     
==========================================
- Hits        40314    40295      -19     
- Misses       1275     1301      +26     

Flag	Coverage Δ
unittests	96.87% <28.57%> (-0.06%)	⬇️

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