Why the scheme matters
The integrator you choose determines whether your simulation is accurate, stable, and physically faithful over long runs. The wrong choice can inject or drain energy that shouldn't change.
Explicit Euler and its limits
Simple and cheap, but only first-order accurate and prone to instability at large timesteps. Fine for quick sketches, risky for anything stiff or long-running.
RK4: the workhorse
Fourth-order Runge–Kutta gives an excellent accuracy-cost balance for non-stiff systems and is the default for most ODE work in PolySim's dynamics engine.
Symplectic integrators
For mechanics, symplectic (energy-preserving) schemes like semi-implicit Euler and Verlet keep orbits and oscillators stable over millions of steps — which is why they power the particle engine.