On Asymptotic Preserving Schemes for a Class of Stochastic Differential Equations in Averaging and Diffusion Approximation Regimes

Splitting integrators for stochastic Lie--Poisson systems

“…On top of that, we study the properties of the stochastic Poisson systems (see Subsection 2.5) and stochastic Poisson integrators (see Subsection 3.3) in a multiscale regime, namely when the Wiener processes are approximated by a smooth noise. The proposed splitting schemes are asymptotic preserving in this diffusion approximation regime, in the sense of the notion recently introduced in [11]. This property, which is not satisfied by standard integrators, is illustrated with numerical experiments.…”

“…Proposition 2 fits in the class of Wong-Zakai approximation results (where a smooth approximation of a Wiener noise leads to a SDE driven by Stratonovich noise). We also refer to [11,Proposition 2.6] for a similar statement (with m = 1), and references therein for ideas of proof. Note that the weak error estimate can be obtained using a variant of the proof of [11,Proposition 2.4], decomposing the error in terms of solutions of Kolmogorov and Poisson equations.…”

“…The definition of effective integrators for the multiscale system (15), which avoid prohibitive time step size restrictions of the type h = O( 2 ), and which lead to consistent discretisation of y(t) when → 0, is a crucial and challenging problem. This question is briefly studied in Section 3.3 below: we define so-called asymptotic preserving schemes (in the spirit of [11]), employing the preservation of the geometric structure satisfied by the stochastic Poisson integrators introduced in the next section.…”

“…The class of numerical methods which satisfy the two requirements above is known as asymptotic preserving schemes. We refer to the recent work [11] where asymptotic preserving schemes were introduced for a class of stochastic differential equations of the type (15). Note that a standard Euler-Maruyama scheme does not satisfy the asymptotic preserving property.…”

“…for all n ≥ 0 and for all fixed h > 0, where y [n] is given by the splitting scheme (18). As a consequence, the scheme ( 24) is an asymptotic preserving scheme, in the sense of [11]: the following diagram commutes…”