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

2022/03/20 by Bréhier, Charles-Edouard
#FOS: Mathematics #Numerical Analysis (math.NA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2203.10600

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 ε vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size Δt vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to ε, in terms of Δt: 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.

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