# Langevin equation

(Difference between revisions)
 Revision as of 17:29, 2 November 2005 (view source)Salva (Talk | contribs)← 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) (3 intermediate revisions not shown) Line 6: Line 6: where $dW(t)$ is a Wiener process. where $dW(t)$ is a Wiener process. - $u'$ is the turbulence intensity and math>  \tau [/itex] ia Lagrangian time-scale. + $u'$ is the turbulence intensity and   \tau [/itex] a Lagrangian time-scale. Th finite difference  approximation of the above equation is Th finite difference  approximation of the above equation is Line 18: Line 18: The Wiener process can be understood as Gaussian random variable with 0 mean The Wiener process can be understood as Gaussian random variable with 0 mean and variance $dt$ and variance $dt$ + + {{Stub}}

## 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 $dt$