Verlet integration

When you must follow an orbiting planet or a box of jostling atoms not for a few steps but for millions, you need an integrator that does not slowly leak or pump energy into the system. Verlet integration is that integrator: a specific way of stepping Newton's equations forward that keeps the total energy bounded over enormously long runs, even though each individual step is not especially accurate. It is the standard tool wherever faithful long-time dynamics of a conservative system matters.

Write the Taylor expansion of the position both forward and backward in time, r(t + h) and r(t - h), and add them. The odd-power terms, the ones that would need the velocity, cancel, leaving r(t + h) = 2 r(t) - r(t - h) + a(t) h^2, where a = F / m is the acceleration. Positions and accelerations are all you need, the local error in position is of order h^4, and, decisively, the scheme is symplectic and time-reversible, so it conserves a nearby 'shadow' energy and shows no secular energy drift. The velocity-Verlet variant, r(t + h) = r(t) + v(t) h + (1/2) a(t) h^2 together with v(t + h) = v(t) + (1/2)[a(t) + a(t + h)] h, supplies synchronized velocities for computing kinetic energy and temperature.

This is precisely why Verlet, not the locally more accurate RK4, is the default integrator of molecular dynamics and gravitational n-body simulation: over long integrations its energy merely oscillates within a fixed band, whereas RK4 slowly drifts away. The honest caveat is that a symplectic method bounds the energy error but does not remove the per-step error, and Verlet is only second-order accurate. It trades peak short-time precision for outstanding long-time stability, which is the right trade for conservative dynamics but the wrong one when you need high accuracy over a short interval.