AD: Extend differentiable programming

The first steps towards differentiating through Trixi were made in #461. The next steps outlined there are
- [x] Benchmark and improve performance of `jacobian_ad_forward` (#464)
- [x] Allow differentiating through a complete solve including time integration (#464)
- [ ] Make callbacks differentiable
- [x] `AnalysisCallback` (#487)
- [ ] `SaveRestartCallback` and `SaveSolutionCallback`: Shall they save the dual numbers or the underlying plain floats (using `Printf.tofloat` in Julia v1.6)? Decision in Trixi meeting: Save the floats for now until we need the more complicated part.
- [ ] `AMRCallback`: Needs improvement of the indicators, see below
- [ ] `VisualizationCallback`: We should probably only visualize the real parts, e.g. using `Printf.tofloat`
- [ ] Fix indicators and shock-capturing volume integrals (issue explained in https://github.com/trixi-framework/Trixi.jl/pull/461#issue-590688249, tracked in https://github.com/trixi-framework/Trixi.jl/issues/1252)
- [ ] Check Euler+gravity
- [ ] Adapt the mesh types to allow differentiating geometric parameters
- [ ] Look for matrix coloring techniques and return sparse matrices to speed-up the computation
- [x] Can we use something like Measurements.jl? (#522)
- [x] Can we use ModelingToolkit.jl?
- Currently not, at least not directly (https://github.com/SciML/ModelingToolkit.jl/issues/929)
- [x] What about other modes of AD, e.g. reverse mode or something like the other tools used in Flux.jl?
- Most of them do not support mutating operations, so they are not really useful for us.
- [ ] What about [Enzyme.jl](https://github.com/wsmoses/Enzyme.jl)? It promises to support mutating operations and works at the LLVM level
- [ ] Check [Tapir.jl](https://juliahub.com/ui/Packages/General/Tapir)
- [ ] Integration with ChainRules to make some parts more efficient, e.g. by using the explicit formulae of Jesse Chan and his student?
- This does currently not work in general since ChainRulesCore does not support mutating operations (https://github.com/JuliaDiff/ChainRulesCore.jl/issues/242). However, it would still be nice to see whether we can get more efficient versions using these explicit formulae.
- Maybe we can provide chain rules for some of the core methods (logarithmic mean, numerical fluxes) to speed up the calculations?