Module integrator 

Time evolution for systems that have no closed form.

Motion is a function of t: ask for the world at 0.7 s and you get it, without having computed 0.6 s first. That is worth a great deal — an exposure can be sampled at seven instants for motion blur, and frame 7 of a recording does not depend on having rendered frame 6 — and it is why drift, oscillation and spin are written the way they are.

It is also not available in general. Three bodies under gravity have no closed form, and neither do contact, heat conduction, or a stiff reaction network. Those systems have to be rolled forward, and frame 7 genuinely does depend on frame 6.

Reproducibility survives the trade, under three rules

• Fixed steps only. Integrator::step takes dt and uses it. An adaptive step chosen from the local error makes the floating-point path depend on the values, so two runs that should agree diverge at the first place one of them decided to halve the step. Where stability demands a smaller step, take a fixed number of substeps — see substeps_for.
• No wall clock. Nothing here reads a timer.
• Ordered reduction. Summing forces in parallel changes the answer, because floating-point addition is not associative. That rule belongs to the domains, but it is the reason State::axpy is a sequential operation on a whole state rather than a per-element one to be farmed out.

Symplectic versus accurate

Integrator::Rk4 is fourth-order accurate and loses energy steadily. velocity_verlet is second-order and does not: its energy error oscillates within a bound instead of drifting, because it preserves the geometric structure of a Newtonian system rather than merely fitting its derivative. Over ten steps RK4 wins; over ten million, it has quietly cooled the system down. For anything conservative — orbits, molecules, an undamped spring — use the symplectic one and let crate::conserved::audit confirm it.

The test module proves exactly this on a harmonic oscillator, against the closed-form energy.

Enums

Integrator

Explicit fixed-step integrators.

Traits

Dynamics

A first-order system: ds/dt = f(s, t).

Newtonian

A Newtonian system: d²x/dt² = a(x, t), with no dependence on velocity.

State

A state vector that an integrator can do arithmetic on.

Functions

substeps_for

How many equal substeps of at most limit it takes to cover dt.

velocity_verlet

One velocity-Verlet step, in place. Symplectic, second order, and the right default for anything whose energy is supposed to stay put.