Integrator Stability Under Frame Jitter — Phase-Space Simulator (2D)

Three Integration Strategies

Explicit Euler — raw jittered dt, no safeguard Symplectic Euler — dt clamped, one step/frame Fixed-timestep RK4 accumulator — correct

Simulated Frame Conditions

Frame jitter (dropped-frame severity) 35%

Simulated FPS cap 60 fps

Oscillator Parameters

Stiffness ω (angular frequency) 3.14 rad/s

Initial amplitude x₀ 1.00

Live Readouts

Frame dt (jittered)0 ms

Explicit Euler energy error0.0%

Symplectic Euler energy error0.0%

RK4 energy error0.0%

RK4 sub-steps this frame0

How it works

A mass on an ideal spring obeys x'' = -ω²x, whose exact solution keeps mechanical energy E = ½v² + ½ω²x² perfectly constant forever. This 2D companion feeds the same jittery, dropped-frame dt stream used by the 3D bouncing-ball sim into three different numerical integrators instead of three different balls, then plots each one's trajectory as a live phase-space portrait (position on the horizontal axis, velocity on the vertical axis) — the shape of that curve is the energy story.

// 1) EXPLICIT (forward) EULER — unconditionally unstable
v -= w2*x*dt; x += v_old*dt;      // spirals outward every single step,
                                   // regardless of how small dt is

// 2) SEMI-IMPLICIT / SYMPLECTIC EULER — bounded, no secular drift
v -= w2*x*dt; x += v_new*dt;      // updates x with the NEW v — this one
                                   // swap keeps the orbit a closed loop

// 3) FIXED-TIMESTEP RK4 ACCUMULATOR — the correct pattern
accumulator += dt;
while (accumulator >= FIXED_DT) { rk4Step(FIXED_DT); accumulator -= FIXED_DT; }
render(state);

Explicit Euler's phase-space orbit is not a closed circle — it is an outward spiral, because the discrete map's determinant is 1+(ωΔt)²>1: every step multiplies the oscillator's energy by that factor, so it grows without bound no matter the frame rate. Symplectic Euler's one-line reordering makes the map area-preserving (determinant exactly 1), so its orbit stays a closed loop with only small bounded wobble — it never drifts away over time. The RK4 accumulator, stepping in small fixed increments completely decoupled from the jittery render clock, keeps energy error negligible and its orbit visually indistinguishable from a perfect circle.

Same lesson as Glenn Fiedler's "Fix Your Timestep!", viewed through a different lens: it is not just whether you clamp dt, but which numerical integrator you pair it with, that decides whether a physics loop is stable.

◤ phase space (x, v)◣ position vs time