Not fully broadcasting methods

[devmotion]

Despite #104 not all methods are fully broadcasting yet. Mostly this is due to JuliaLang/julia#22255.

Methods that are not fully broadcasting (usually both not in-place and in-place versions if existent):

• Feagin10
• Feagin12
• Feagin14
• RK4
• TanYam7
• DP8
• TsitPap8
• OwrenZen4
• OwrenZen5
• BS5
• Tsit5
• DP5
• GenericImplicitEuler
• GenericTrapezoid (only in-place)
• ImplicitEuler
• Trapezoid
• TRBDF2
• Hairer4/Hairer42 (only in-place)
• KenCarp4 (only in-place)
• Kvaerno5 (only in-place)
• KenCarp5 (only in-place)
• IRKN3
• IRKN4
• DPRKN6
• DPRKN8
• DPRKN12
• ERKN4
• Nystrom5VelocityIndependent
• Rosenbrock23 (only in-place)
• Rosenbrock32 (only in-place)
• Rosenbrock33 (only in-place)
• Rosenbrock34 (only in-place)
• Rosenbrock4 (only in-place)
• Rodas4 (only in-place)
• Rosenbrock5
• Vern6
• Vern7
• Vern8
• Vern9

DP5Threaded is by design not broadcasting.

Moreover, we have to assure that also corresponding interpolation methods are broadcasting.

[devmotion]

We also have to get rid of lines such as https://github.com/JuliaDiffEq/OrdinaryDiffEq.jl/blob/master/src/integrators/rosenbrock_integrators.jl#L55-L57 in order to get fully broadcasting methods. Just replacing them with @. W = mass_matrix - γ*J does not work (see #104 (comment)). Maybe we can use something like

if typeof(mass_matrix) <: UniformScaling
    W .= mass_matrix - dt*J
else
    @. W = mass_matrix - dt*J
end

?

[ChrisRackauckas]

I think there are some Runge-Kutta Nystrom methods in this list as well.

[ChrisRackauckas]

Maybe JuliaLang/julia#23912 fixes JuliaLang/julia#22255?

[ChrisRackauckas]

JuliaLang/julia#22255 is fixed on Julia master!

[YingboMa]

Sadly, we still need to wait for the fix for JuliaLang/julia#28126.

[YingboMa]

With JuliaLang/julia#30973, I got

julia> @btime solve($(ODEProblem((du,u,p,t)->@.(du=0.01*u), ones(100), (0,100.))), Tsit5(), save_on=false); # loop
  13.446 μs (100 allocations: 28.83 KiB)

julia> @btime solve($(ODEProblem((du,u,p,t)->@.(du=0.01*u), ones(100), (0,100.))), Tsit5(), save_on=false); # broadcast
  13.075 μs (100 allocations: 28.83 KiB)

🎉