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We introduce and study a notion of Asymptotic Preserving schemes, related toconvergence in distribution, for a class of slow-fast Stochastic DifferentialEquations. In some examples, crude schemes fail to capture the correct limitingequation resulting from averaging and diffusion approximation procedures. Wepropose examples of Asymptotic Preserving schemes: when the time-scaleseparation vanishes, one obtains a limiting scheme, which is shown to beconsistent in distribution with the limiting Stochastic Differential Equation.Numerical experiments illustrate the importance of the proposed AsymptoticPreserving schemes for several examples. In addition, in the averaging regime,error estimates are obtained and the proposed scheme is proved to be uniformlyaccurate.

On Asymptotic Preserving schemes for a class of Stochastic Differential Equations in averaging and diffusion approximation regimes