Integrator Race: Euler vs Verlet vs RK4

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Step size Δt 0.040

Gravity strength GM 1.00

Initial eccentricity 0.45

Integrators shown

Explicit Euler Symplectic Euler Velocity Verlet RK4

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Integrator	Energy drift

Sim time

0.0 s

Steps taken

How it works

All four bodies orbit the same point mass under F = -GMm·r̂/r² and start from the identical position and velocity. Each is advanced by a different numerical integrator using the same step size Δt, so any difference you see is caused purely by the integration method:

Explicit Euler:      x' = x + v·Δt            v' = v + a(x)·Δt
Symplectic Euler:     v' = v + a(x)·Δt         x' = x + v'·Δt
Velocity Verlet:      x' = x + v·Δt + ½a(x)Δt²  v' = v + ½[a(x)+a(x')]·Δt
RK4:                  weighted average of 4 slope evaluations per step

Specific orbital energy E = ½|v|² − GM/r should stay constant on a closed orbit. Explicit Euler is not symplectic: it systematically adds energy every step, so its orbit visibly spirals outward and its drift readout grows without bound. Symplectic (semi-implicit) Euler and Velocity Verlet are symplectic integrators — their energy oscillates but never drifts on average, so the orbit stays closed even at a coarse Δt. RK4 is far more accurate per step than any of the first-order methods but is not symplectic either, so at a large enough Δt its energy drift also creeps up, just much more slowly than plain Euler's.

Δt slider — the physics step size shared by all four integrators; push it up to exaggerate the divergence.
GM slider — gravitational parameter of the central mass; higher GM curves orbits faster for a given Δt, which stresses Euler harder.
Eccentricity slider — sets the initial tangential speed below/above circular, producing an elliptical orbit; the fast perihelion pass is where Euler-family error accumulates fastest.
Reset orbits — reseeds all four bodies at the same initial state so you can compare a fresh run at the current settings.

This is exactly the trade-off real orbital-mechanics and rigid-body engines face: symplectic integrators (Verlet-family) are the default in long-running N-body and game-physics solvers precisely because they don't leak or gain energy over millions of steps, even though a single RK4 step is more accurate.

Central mass Explicit Euler Symplectic Euler Velocity Verlet RK4

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