Verlet Integration - Velocity Verlet

A related, and more commonly used, algorithm is the Velocity Verlet algorithm, similar to the leapfrog method, except that the velocity and position are calculated at the same value of the time variable (Leapfrog does not, as the name suggests). This uses a similar approach but explicitly incorporates velocity, solving the first-timestep problem in the Basic Verlet algorithm:

It can be shown that the error on the Velocity Verlet is of the same order as the Basic Verlet. Note that the Velocity algorithm is not necessarily more memory consuming, because it's not necessary to keep track of the velocity at every timestep during the simulation. The standard implementation scheme of this algorithm is:

1. Calculate:
2. Calculate:
3. Derive from the interaction potential using
4. Calculate: .

Eliminating the half-step velocity, this algorithm may be shortened to

1. Calculate:
2. Derive from the interaction potential using
3. Calculate: .

Note, however, that this algorithm assumes that acceleration only depends on position, and does not depend on velocity .

One might note that the long-term results of Velocity Verlet, and similarly of Leapfrog are one order better than the semi-implicit Euler method. The algorithms are almost identical up to a shifted by half of a timestep in the velocity. This is easily proven by rotating the above loop to start at Step 3 and then noticing that the acceleration term in Step 1 could be eliminated by combining Steps 2 and 4. The only difference is that the midpoint velocity in velocity Verlet is considered the final velocity in semi-implicit Euler method.

The global error of all Euler methods is of order one, whereas the global error of this method is, similar to the midpoint method, of order two. Additionally, if the acceleration indeed results from the forces in a conservative mechanical or Hamiltonian system, the energy of the approximation essentially oscillates around the constant energy of the exactly solved system, with a global error bound again of order one for semi-explicit Euler and order two for Verlet-leapfrog. The same goes for all other conservered quantities of the system like linear or angular momentum, that are always preserved or nearly preserved in a symplectic integrator.