Geometric Ergodicity and Optimal Error Estimates for a Class of Novel Tamed Schemes to Super-linear Stochastic PDEs

TL;DR

This paper introduces new tamed numerical schemes for super-linear stochastic PDEs that preserve stability and ergodicity, achieving optimal convergence rates in both multiplicative and additive noise scenarios.

Contribution

The authors develop a novel class of tamed schemes that maintain Lyapunov functionals and inherit geometric ergodicity for super-linear SPDEs, with proven optimal error estimates.

Findings

Schemes preserve Lyapunov functionals and stability.

Galerk-in-based schemes inherit ergodicity.

Achieve optimal strong convergence rates.

Abstract

We construct a class of novel tamed schemes that can preserve the original Lyapunov functional for super-linear stochastic PDEs (SPDEs), including the stochastic Allen--Cahn equation, driven by multiplicative or additive noise, and provide a rigorous analysis of their long-time unconditional stability. We also show that the corresponding Galerkin-based fully discrete tamed schemes inherit the geometric ergodicity of the SPDEs and establish their convergence towards the SPDEs with optimal strong rates in both the multiplicative and additive noise cases.