Verlet integration

It is a variant of the Verlet integration method.

Then, using that density, forces on the nuclei can be computed, to update the trajectories (using, e.g. the Verlet integration algorithm).

The most notable thing that is now easier due to using Verlet integration rather than Eulerian is that constraints between particles are very easy to do.

Using only the predictor formula and the corrector for the velocities one obtains a direct or explicit method which is a variant of the Verlet integration method:

The matrix code can be reused: The dependency of the forces on the positions can be approximated locally to first order, and the verlet integration can be made more implicit.

Where Euler's Method uses the Forward difference approximation to the first derivative, Verlet Integration can be seen as using the Central difference approximation to the second derivative:

An iterative process is followed by simulating a pseudo-dynamic process in time, with each iteration based on an update of the geometry, similar to Leapfrog integration and related to Velocity Verlet integration.

Verlet integration: used by Hitman: Codename 47 and popularized by Thomas Jakobsen, this technique models each character bone as a point connected to an arbitrary number of other points via simple constraints.

Verlet integration improves the accuracy of the integration to within fourth-order Taylor series terms, and the Runge-Kutta method which is gaining popularity further improves this accuracy to within fifth-order Taylor series terms.

For big matrices sophisticated solvers (look especially for "The sizes of these small dense matrices can be tuned to match the sweet spot" in [1]) for sparse matrices exist, any self made Verlet integration has to compete with these.

Verlet integration was used by Carl Størmer to compute the trajectories of particles moving in a magnetic field (hence it is also called Störmer's method) and was popularized in molecular dynamics by French physicist Loup Verlet in 1967.

The Verlet integration would automatically handle the velocity imparted via the collision in the latter case, however note that this is not guaranteed to do so in a way that is consistent with collision physics (that is, changes in momentum are not guaranteed to be realistic).

In a famous 1967 paper he used what is now known as Verlet integration (a method for the numerical integration of equations of motion) and the Verlet list (a data structure that keeps track of each molecule's immediate neighbors in order to speed computer calculations of molecule to molecule interactions).