Numerical Techniques for Multi-Scale Dynamical Systems with Stochastic Effects

Numerical schemes are presented for dynamical systems with multiple time-scales. Two classes of methods are discussed, depending on the time interval on which the evolution of the slow variables in the system is sought. On rather short time intervals, the slow variables satisfy ordinary differential equations. On longer time intervals, however, fluctuations become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using short simulations of appropriate auxiliary systems.