EXPO 2026: "Euler-Cromer Time-Stepping as Alternating Projections for Second-Order ODEs"

Описание: Sat | 12:30 PM–1:00 PM ET
Carey Witkov; Physics Department, MIT, Cambridge, MA USA

Abstract: Euler-Cromer (and leapfrog/Verlet algorithms) are used in ODE modeling because they often give qualitatively correct phase portraits even at modest step sizes, especially in oscillatory and energy-based systems. This talk gives a geometric explanation: for standard linear models, Euler-Cromer can be interpreted as alternating projections onto affine (hence convex) constraint sets in the phase plane. For more general (e.g., nonlinear) models, even when the relevant constraint sets are not convex, the idea of alternating enforcement of constraints is preserved.
Starting from the first-order system: x′(t)=v(t), v′(t)=a(x,v,t), one substep enforces a dynamical constraint (update v so it is consistent with the modeled acceleration) and the other enforces a kinematic constraint (update x so it is consistent with the updated velocity). Each substep acts like a projection onto a constraint set, so a full timestep becomes an alternating-projection map. This view clarifies why update order matters, why Euler-Cromer differs qualitatively from forward Euler, and why certain invariants (or near-invariants) are better respected in practice.
Examples (simple harmonic oscillator; driven/damped oscillator) show how phase portraits and energy behavior follow from successive constraint enforcement rather than from numerical correctors. Classroom-ready diagnostics such as residual plots, phase-space drift, and step-size sweeps connect modeling assumptions directly to numerical behavior.

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