Langevin simulation provides an effective way to study collisional effects in
beams by reducing the six-dimensional Fokker-Planck equation to a group of
stochastic ordinary differential equations. These resulting equations usually
have multiplicative noise since the diffusion coefficients in these equations
are functions of position and time. Conventional algorithms, e.g. Euler and
Heun, give only first order convergence of moments in a finite time interval.
In this paper, a stochastic leap-frog algorithm for the numerical integration
of Langevin stochastic differential equations with multiplicative noise is
proposed and tested. The algorithm has a second-order convergence of moments in
a finite time interval and requires the sampling of only one uniformly
distributed random variable per time step. As an example, we apply the new
algorithm to the study of a mechanical oscillator with multiplicative noise.Comment: 3 pages, 4 figures, to submit to XX International LINAC conferenc