AD on callback directionality?

[ChrisRackauckas]

Gets one more bounce out of:

using Markdown
using InteractiveUtils
using ForwardDiff

# ╔═╡ 2d729750-5455-11eb-323b-85261dc4f76a
begin # define independent package environment
	import Pkg
	Pkg.activate(mktempdir())
	Pkg.add("Plots"); using Plots
	Pkg.add("DifferentialEquations"); using DifferentialEquations
end

# ╔═╡ 59df1e80-5455-11eb-2323-1b07750a4d6b
function f(du,u,p,t)
  du[1] = u[2]
  du[2] = -p
end

# ╔═╡ 68dbca00-5455-11eb-36ba-ad0f2db6b034
function condition(u,t,integrator) # Event when event_f(u,t) == 0
  u[1]
end

# ╔═╡ 6ea96280-5455-11eb-07d4-adf2ce716897
function affect!(integrator)
  println("Bounce at t = ",integrator.t)
  integrator.u[2] = -0.1*integrator.u[2]
end

# ╔═╡ 7401cb00-5455-11eb-0b6d-973b7ed34eee
cb = ContinuousCallback(condition,affect!,interp_points=10000)

# ╔═╡ 7cdd3fc0-5455-11eb-15a2-07bf5923d0f7
function sim()
	u0 = [50.0,0.0]
	tspan = (0.0,5.0)
	p = 9.8
	prob = ODEProblem(f,u0,tspan,p)
	solve(prob,Tsit5(),callback=cb,dtmax=1)
end

# ╔═╡ 2e6ff0ce-545a-11eb-3537-251e4eff7339
sol = sim()

# ╔═╡ 790adb00-5455-11eb-301c-61f7a5f0d3cf
function plotsol(tspan, title)
	p1 = plot(sol, vars=[1], tspan=tspan, 
		framestyle=:zerolines, ylabel="position", title=title)
	p2 = plot(sol, vars=[2], tspan=tspan,
		framestyle=:zerolines, ylabel="velocity")
	plot(p1,p2,layout=(2,1),link=:x, leg=false)
end

# ╔═╡ 63237f80-545b-11eb-37a5-830fefdc5c96
plotsol((0, 5), "can see first and second bounces clearly")

# ╔═╡ a830ce20-545b-11eb-3e3c-694bb6408666
plotsol((3.83,3.91), "last four bounce are in here")

# ╔═╡ bb97b18e-545b-11eb-0e1f-ebf59b70e16f
plotsol((3.895, 3.905), "last three bounces")

# ╔═╡ 596d1ea0-545c-11eb-07e3-3fcef590c7bd
plotsol((3.903, 3.905), "last two bounces")

# ╔═╡ cb3366c0-545c-11eb-2345-571ee3526ba5
plotsol((3.90415, 3.9043), "last bounce")

https://discourse.julialang.org/t/diffeq-tolerances-on-continuouscallbacks/53181

[ChrisRackauckas]

This is more of an experimental branch, not meant to merge. Two things to note:

1. This requires allocating from the interpolation since there's no cache with duals
2. It would add more dependencies.