Adapt jacobian_ad_forward for hyperbolic-parabolic semidiscretizations

[DanielDoehring]

Related to #1147.

[codecov]

Codecov Report

Merging #1589 (47cc392) into main (ddf0892) will decrease coverage by 2.73%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##             main    #1589      +/-   ##
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- Coverage   85.63%   82.90%   -2.73%     
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  Files         401      402       +1     
  Lines       33004    33052      +48     
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- Hits        28260    27399     -861     
- Misses       4744     5653     +909     

Flag	Coverage Δ
unittests	82.90% <100.00%> (-2.73%)	⬇️

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Files Changed	Coverage Δ
...d_dgsem/elixir_navierstokes_taylor_green_vortex.jl	100.00% <ø> (ø)
...d_dgsem/elixir_navierstokes_taylor_green_vortex.jl	100.00% <100.00%> (ø)
...ization/semidiscretization_hyperbolic_parabolic.jl	98.97% <100.00%> (+2.34%)	⬆️

... and 138 files with indirect coverage changes

[ranocha]

Thanks! Could you please post some plots of the spectra computed for linear advection diffusion? Do the Jacobians agree well with what you get from linear_structure?

[DanielDoehring]

Thanks! Could you please post some plots of the spectra computed for linear advection diffusion?

Sure, for 1D, DGSEM(3), Local Lax-Friedrichs, Domain $[- \pi, \pi]$ and 128 elements I attach 5 ratios $a/d = 10^n, n = 0, \dots, 4$.
If the grid is refined, the diffusive part starts to become more influential since then $\frac{d}{\Delta x^2}$ grows quicker than $\frac{a}{\Delta x}$.

Spectra

Do the Jacobians agree well with what you get from linear_structure?

I thought about implementing linear_structure as well for parabolic semidiscretizations but since I do not need it and not really understand what is going on here

Trixi.jl/src/semidiscretization/semidiscretization.jl

Lines 170 to 177 in e1e680c

	b_tmp = copy(b)
	# wrap the linear operator
	A = LinearMap(length(u_ode), ismutating = true) do dest, src
	rhs!(dest, src, semi, t0)
	@. dest += b_tmp
	dest
	end

decided to omit it for the moment.

[ranocha]

Sure

Thanks

[ranocha]

Thanks!

[ranocha]

macOS MPI CI seems to time out again - see #1590. I will merge this manually.