Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

TL;DR

This paper establishes uniform weak error estimates for an asymptotic preserving scheme applied to slow-fast parabolic semilinear SPDEs, ensuring accuracy across different time scales and extending previous finite-dimensional results.

Contribution

It provides the first uniform weak error analysis for an asymptotic preserving scheme in an infinite-dimensional setting for SPDEs with multiple time scales.

Findings

The scheme achieves uniform accuracy with respect to the scale parameter .

The analysis extends to infinite-dimensional Kolmogorov equations.

The fast component discretization uses a modified Euler scheme for SPDEs.

Abstract

We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $\epsilon$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $\Delta t$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $\epsilon$, in terms of $\Delta t$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in an infinite dimensional framework. The fast component is discretized using the modified Euler scheme for SPDEs introduced in a recent work. Proving the weak error estimates requires delicate analysis of the regularity properties of solutions of infinite dimensional Kolmogorov equations.