Interaction between Adapt.jl and jacobian_ad

[vchuravy]

The semi itself is constructed with two arguments, RealT and uEltype, that determine the type of internal caches, etc. By default, they will be Float64, but there should be some clear interaction between all these types.

For example, how would you see the interaction with AD like in

Trixi.jl/src/semidiscretization/semidiscretization.jl

Lines 268 to 278 in 36dab43

	function _jacobian_ad_forward(semi, t0, u0_ode, du_ode, config)
	new_semi = remake(semi, uEltype = eltype(config))
	# Create anonymous function passed as first argument to `ForwardDiff.jacobian` to match
	# `ForwardDiff.jacobian(f!, y::AbstractArray, x::AbstractArray,
	# cfg::JacobianConfig = JacobianConfig(f!, y, x), check=Val{true}())`
	J = ForwardDiff.jacobian(du_ode, u0_ode, config) do du_ode, u_ode
	Trixi.rhs!(du_ode, u_ode, new_semi, t0)
	end
	return J
	end

Originally posted by @ranocha in #2212 (comment)