ODE Integrator Race — Euler vs RK2 vs RK4

Integrators

Euler (1st order)0.0%

Midpoint (RK2)0.0%

Classical RK40.0%

Parameters

Step size h 0.040

Eccentricity e 0.50

Steps per frame 4

Live Readouts

Sim time t

0.00

Steps taken

Worst energy drift

0.0%

How it works

All three integrators solve the same 2-body gravity ODE, starting from the identical initial state (r₀, v₀) on a Kepler ellipse with semi-major axis a and eccentricity e:

d²r/dt² = -GM · r / |r|³
Energy:  E = ½|v|² - GM/|r|   (conserved exactly by the true orbit)

Each method advances the state by one step h differently:

Euler:    r,v += h·f(r,v)                         → error O(h) per step
Midpoint: k = f(r + h/2·v, ...) at half-step        → error O(h²) per step
RK4:      weighted average of 4 slope evaluations   → error O(h⁴) per step

Step size h — larger steps make truncation error explode fastest for Euler, slower for RK2, and barely visible for RK4 until h gets large.
Eccentricity e — a more elongated ellipse swings through perihelion faster, stressing the fixed step size exactly where curvature is highest.
Energy drift — since the true orbit conserves energy exactly, (E(t) − E₀)/|E₀| is a direct, visible measure of each method's accumulated numerical error; Euler drifts monotonically (often gaining energy and spiraling outward), while RK4 stays flat far longer for the same h.

This is the same trade-off every real ODE solver (orbital mechanics, circuit simulation, molecular dynamics) has to make between step size, computational cost and long-term accuracy.

Euler Midpoint (RK2) RK4 True ellipse

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