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Need some help with Verlet algorithm

  1. Feb 26, 2008 #1
    1. The problem statement, all variables and given/known data

    I am gonna use the velocity verlet algorithm to simulate the position, velocity and acceleration at time t of some particle.

    2. Relevant equations

    We got the Langevin equation: a(t) = -v(t)/tau - U'(x) / m + F_f(t) / m

    Where tau is the mass divided by the friction coefficient, U'(x(t)) the external force and F_f(t) the fluctuating force. Let's say the external force is harmonic, ½kx(t)^2, so U'(x(t)) / m = kx(t)/m.

    I have so far typed in x(t+dt), v(t+dt/2) and I need to find what a(t+dt) is. It's easier if you go see http://en.wikipedia.org/wiki/Verlet_integration for the equations i refer to :)

    As for a(t+dt) I am in doubt how I should find this, what I tried so far is:

    a(t+dt) = -1/m * dV(x(t+dt))/dx

    = -v(t) / tau - k*x(t+dt) / m + F_f(t)

    (Where I take the value of the fluctuating force randomly from a normaldistribution)

    My doubt is about the use of v(t), can I use v(t) in the above equation? I don't see how many other options I have since we need a(t+dt) to be able to calculate v(t+dt), or have I missed something?

    I hope someone can help me a bit, if you need more info or anything is unclear, please tell me here.
     
  2. jcsd
  3. Feb 27, 2008 #2
    No one able to help? :)
     
  4. Feb 27, 2008 #3
    The verlet integrator is semi-implicit, which means exactly what you've noticed: you need [tex]v(t+dt)[/tex] to calculate [tex]a(t+dt)[/tex]. In general this means that you need to solve an equation at each step (numerically). Do you have to use Verlet? The usual Runge-Kutta family is pretty good, or even just plain ol' midpoint rule?
     
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