# Langevin equation

(Difference between revisions)
 Revision as of 12:20, 15 June 2007 (view source)YlzEnr (Talk | contribs)m← Older edit Latest revision as of 14:30, 15 June 2007 (view source)Jola (Talk | contribs) m (Reverted edits by YlzEnr (Talk); changed back to last version by Salva) Line 2: Line 2: the Langevin equation is the Langevin equation is :$:[itex] - dU(t) = - U(t) \frac{dt}{\tau} \frac{2 u'}{\tau}^{1/2} dW(t) + dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)$ [/itex] Line 11: Line 11: :$:[itex] - U(t \Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N} + U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}$ [/itex]

## Latest revision as of 14:30, 15 June 2007

The stochastic differential equation (SDE) for velocity component $U(t)$, the Langevin equation is

$dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)$

where $dW(t)$ is a Wiener process. $u'$ is the turbulence intensity and $\tau$ a Lagrangian time-scale.

Th finite difference approximation of the above equation is

$U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}$

where $\mathcal{N}$ is a standardized Gaussian random variable with 0 mean an unity variance which is independent of $U$ on all other time steps (Pope 1994). The Wiener process can be understood as Gaussian random variable with 0 mean and variance $dt$