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How you choose the timestep is the physics
Four integrators on identical initial conditions. The energy trace below the field is the subject — not how large the error is, but whether it grows.
Starting…
- |ΔE/E|
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- worst |ΔE/E|
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- Q = T/|U|
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- elapsed
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- bound
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- force evals / frame
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What the trace shows that the number cannot
Pick Two-body orbit and watch the energy trace under the field. OnSymmetric it settles into a band and stays there however long you run. Switch to Hermite — the same order, the same corrector, the same softening — and the trace climbs. Neither scheme is less accurate per step. The difference is entirely in how the step size is chosen.
Hermite sizes its step from the state it is leaving, dt = h(ξ₀). A map whose step depends on only one of its two endpoints is not invariant under reversing time, and so has no reason to give the energy back. The symmetric scheme takesdt = ½[h(ξ₀) + h(ξ₁)] — a rule that reads the same forwards and backwards — and pays for it by having to solve a fixed point, because ξ₁ depends on the step that depends on ξ₁.
Press Reverse on the two-body orbit. The velocities flip and the run retraces itself; the symmetric scheme returns to where it began, and the others do not. That is the property, and everything else on this page is downstream of it.
Measured
| orbits | symmetric | asymmetric |
|---|---|---|
| 4 | 2.38e−5 | 2.28e−4 |
| 16 | 2.39e−5 | 7.77e−4 |
| 64 | 2.43e−5 | 2.96e−3 |
| 256 | 2.56e−5 | 1.16e−2 |
Flat to 7% against a factor of 51. The cost is real and runs the other way: one symmetric step is eight force evaluations against one, so on a 200-star cluster it conserves five orders better and takes 39× the work — 147 806 force evaluations per crossing time against 3758, and 222 seconds of wall clock against 7. That is why it is the default for two bodies and an opt-in for a cluster, and why the scenario decides rather than the reader.
Softening, on the cluster, is scaffolding for the fixed-step schemes rather than physics. Take it to zero and watch FSI4 and the leapfrog come apart while the two adaptive schemes carry on: an adaptive step is the mechanism that resolves a close pair, and softening is the substitute a fixed step is forced to use instead.