# How does Velocity Verlet integration improve accuracy in modeling fast dynamics?

• squid
In summary, the conversation discussed the concept of Velocity Verlet integration and its differences from Verlet integration. It was mentioned that Velocity Verlet integration yields higher accuracy for velocity compared to plain Verlet integration. The conversation also touched upon the use of higher-order propagation techniques and the potential benefits of alternative techniques such as Beeman's algorithm for modeling fast dynamics.
squid
Hi!
Could anyone explain me why Velocity Verlet integration works and how did Loup Verlet come up with it?

Thanks!

I saw that too. Links from that page suggest that "Velocity Verlet" integration is somewhat different from "Verlet integration".

EDIT

Found it http://www.ch.embnet.org/MD_tutorial/pages/MD.Part1.html". Velocity Verlet integration integrates velocity as well as position via a modified Euler scheme:
$$v(t+\Delta t) = v(t) + \frac 1 2 (a(t)+a(t+\Delta t))\Delta t$$

Plain Jane Verlet integration computes velocity post-integration, resulting in $$O(\Delta t^2)$$ velocity errors. The Velocity Verlet integration yields $$O(\Delta t^3)$$ accuracy for velocity.

/EDIT

We typically use higher-order propagation techniques to achieve a high level of accuracy. It's pretty hard to beat good old RK4 in a regime where the integration frequency has to match the thruster control frequency (10 to 100 Hz or so) while the orbital dynamics operate at a much slower frequency.

However, we sometimes need to revert to lower order techniques to model flex (very fast dynamics). This technique and related ones (e.g., http://en.wikipedia.org/wiki/Beeman%27s_algorithm" ) look very promising.

Thanks to the OP.

Last edited by a moderator:

## 1. What is Velocity Verlet integration?

Velocity Verlet integration is a numerical integration method used to solve ordinary differential equations (ODEs) in physics and other fields. It is a variant of the Verlet algorithm and is commonly used in molecular dynamics simulations.

## 2. How does Velocity Verlet integration work?

Velocity Verlet integration works by calculating the position and velocity of a system at a given time step based on the previous time step and the current acceleration. It uses a Taylor series expansion to approximate the position and velocity at the next time step.

## 3. What are the advantages of using Velocity Verlet integration?

Velocity Verlet integration is a symplectic integrator, which means it conserves energy and momentum in a system. It is also more accurate and stable than other integration methods, such as the Euler method, especially for systems with varying forces.

## 4. What are the limitations of Velocity Verlet integration?

One limitation of Velocity Verlet integration is that it requires the position and velocity of a system at the previous time step, which can make it computationally expensive for large systems or simulations with small time steps. It also may not accurately capture the behavior of systems with high-frequency oscillations.

## 5. What are some applications of Velocity Verlet integration?

Velocity Verlet integration is commonly used in molecular dynamics simulations to study the motion and behavior of particles, such as atoms and molecules. It is also used in other fields, such as astrophysics, to model the dynamics of celestial bodies and in computer graphics to simulate the movement of objects.

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