How wrong is a simulator's pendulum period at its default timestep?
research/free-integperiod
How wrong is a simulator's pendulum period at its default timestep? Exact elliptic-integral period against Euler, semi-implicit Euler, Verlet and RK4: Euler fails (14% period error at 2 ms), the symplectic pair is 0.75 ppm, and releasing from rest hides the symplectic error.
What it found
Exact large-angle period (complete elliptic integral, 2.367842 s at 90°) against explicit Euler, semi-implicit Euler, velocity Verlet and RK4 at 0.5 to 16 ms. Explicit Euler is unusable (energy +81% and period +14% after 60 s at 2 ms). Semi-implicit Euler and Verlet keep energy error under 0.3% and have period error 0.75 ppm at 2 ms, falling 4× per halving; RK4 reaches 1.8×10−6 ppm. Started on the same orbit the two second-order methods have identical period error, but releasing from rest makes semi-implicit Euler look 4th order (≤10−4 ppm), an initial-condition artefact. Textbook integrators on an undamped pendulum, not the MuJoCo or PyBullet code. Every input is sourced or swept; nothing is assumed.
A figure from its own output
Paper, code and results
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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.