Handles restarting problem of adaptive time integration methods

[ArseniyKholod]

If time integration method uses time step controller like PID controller, than to obtain an exact solution after restarting the computation we have to save some additional data in SaveRestartCallback. This is a first shot.

[codecov]

Codecov Report

Patch coverage: 91.30% and project coverage change: -6.60% ⚠️

Comparison is base (92dedde) 92.21% compared to head (a41f45f) 85.61%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1565      +/-   ##
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- Coverage   92.21%   85.61%   -6.60%     
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  Files         418      418              
  Lines       34223    34203      -20     
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- Hits        31558    29281    -2277     
- Misses       2665     4922    +2257     

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unittests	85.61% <91.30%> (-6.60%)	⬇️

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Files Changed	Coverage Δ
src/Trixi.jl	43.48% <ø> (ø)
src/callbacks_step/save_restart_dg.jl	25.73% <90.00%> (-35.14%)	⬇️
src/callbacks_step/save_restart.jl	91.67% <92.31%> (+0.14%)	⬆️

... and 65 files with indirect coverage changes

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[ArseniyKholod]

General concept is, that we have to save some data for integrator.controller, to obtain exactly the same solution as without restart. So I added additional functions that saves them in SaveRestartCallback. Problem in a restarting file is that we have to load these data after creating an integrator, that occurs inside solve() function. It's the reason why I decided to implement a loading stuff inside new callback LoadRestartCallback. While initializing this callback we load all data for integrator.controller. After initializing this callback is never called, because function that activate callback returns always false.

As was said in a discussion in trixi_students I tested that implementation for these solvers for serial and parallel execution:
All with adaptive timestep from Stability-Preserving-Runge-Kutta-Methods-for-Hyperbolic-PDEs:

• SSPRK43()
• SSPRK432()

And some with adaptive timestep from Low storage methods:

• CKLLSRK85_4FM_4R
• RDPK3Sp35
• RDPK3SpFSAL35
• RDPK3SpFSAL49

All of tested methods works correctly with new callback, and works either with some additional L_inf, L2 error or unstable without it.

[ArseniyKholod]

@SimonCan, what do you think about this idea?
Of coarse I'll add some asserts a bit later and other checks.
Now implementation is valid only for PIDController, PIController and IController, others will not produce an error, but also solutions with and without restart will slightly differ. I'll add a detailed support for other controllers, if it makes sence.

Example of using new callback you can find above.

[SimonCan]

Looks great.
Could you add example elixirs like the ones in your comments? I was thinking of asimilar design as in examples/structured_2d_dgsem/elixir_advection_restart.jl where they call first call elixir_advection_extended.jl and then load the data for a restart.

[ArseniyKholod]

Michael, @sloede, may be you have some quick ideas or already saw such problem, why this error could appear? Something wrong with allocations. But error appeared first time when I added only some comments, without changing a code, strange, I'm completely at a loss.
https://github.com/trixi-framework/Trixi.jl/actions/runs/5977210124/job/16216690034?pr=1565#step:7:3322
The same error I got in my new PR (#1614), these PRs seem to have nothing in common.
https://github.com/trixi-framework/Trixi.jl/actions/runs/5977692959/job/16218213885?pr=1614#step:7:3322
One more failing test: https://github.com/trixi-framework/Trixi.jl/actions/runs/5977692959/job/16218213284?pr=1614#step:7:8346

[sloede]

LGTM! It would be good to also get a review from @ranocha

[ranocha]

Did you add a test using a PIDController?

[ArseniyKholod]

Did you add a test using a PIDController?

Yes, in test_mpi_tree.jl with RDPK3SpFSAL49 that uses PIDController by default. And code part, that is used only by PIDController, is covered by coverage test