Optimize VerletLeapfrog method

[efaulhaber]

Checklist

• Appropriate tests were added (does not require additional tests, method is already tested)
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated (no docs changes necessary, method and API didn't change)
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API (no new docs)

Additional context

The VerletLeapfrog method is currently using the generic Symplectic2Cache implementation:

OrdinaryDiffEq.jl/lib/OrdinaryDiffEqSymplecticRK/src/symplectic_perform_step.jl

Lines 222 to 246 in 4b29bab

	@muladd function perform_step!(integrator, cache::Symplectic2Cache, repeat_step = false)
	@unpack t, dt, f, p = integrator
	@unpack a1, a2, b1, b2 = cache.tab
	duprev, uprev, _, kuprev = load_symp_state(integrator)
	du, u, kdu, ku = alloc_symp_state(integrator)
	# update position
	@.. broadcast=false u=uprev + dt * b1 * kuprev
	# update velocity
	f.f1(kdu, duprev, u, p, t)
	@.. broadcast=false du=duprev + dt * a1 * kdu
	# update position & velocity
	tnew = t + a1 * dt
	f.f2(ku, du, u, p, tnew)
	@.. broadcast=false u=u + dt * b2 * ku
	f.f1(kdu, du, u, p, tnew)
	@.. broadcast=false du=du + dt * a2 * kdu
	f.f1(kdu, du, u, p, tnew)
	f.f2(ku, du, u, p, tnew)
	OrdinaryDiffEqCore.increment_nf!(integrator.stats, 3)
	integrator.stats.nf2 += 2
	store_symp_state!(integrator, cache, kdu, ku)
	end

This contains 3 calls to f1 and 2 calls to f2. The kick-drift-kick scheme of the Leapfrog method only requires a single call to f1 and a single call to f2 in each step, which is the main selling point of the method, making the current implementation basically useless.

I implemented an optimized step for this method, following the VelocityVerlet method, which also has its own step function.

[ChrisRackauckas]

Yeah that looks right, seems it just got over simplified.

The interpolation should be specialized though so that it goes to linear instead of Hermite then, since the derivative would be off. Or a special centered Hermite would need to be used.

[efaulhaber]

To be honest, I have no idea what I need to change to make this use a linear interpolation. Could you point me to the code that I need to change?

[ChrisRackauckas]

I just realized we were doing that a bit bespoke. Set this branch up on top of #2560 with the trait and I think it's good to go.

[efaulhaber]

Like so?

[efaulhaber]

Note that this was a bug with the VelocityVerlet method.

[ChrisRackauckas]

Rebase onto the latest master and I think this is good to go.

[ChrisRackauckas]

Test fails.