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

Zhihui Liu, Jie Shen

TL;DR

The paper develops a class of nonlinearly taming numerical schemes for super-linear SPDEs, preserving the Lyapunov structure and achieving unconditional long-time stability. It proves that TEM and Galerkin-based discretizations inherit geometric ergodicity under nondegeneracy and derives strong error bounds in the multiplicative-noise setting, with optimal convergence rates for additive noise. The results cover stochastic Allen–Cahn dynamics in dimensions up to three and provide high-order convergence guarantees under strengthened regularity, offering practical, nonlinearity-explicit integrators for ergodic SPDEs. The method delivers efficient, provably accurate long-time simulations and invariant-measure approximations for SPDEs with non-Lipschitz nonlinearities.

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.

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

TL;DR

The paper develops a class of nonlinearly taming numerical schemes for super-linear SPDEs, preserving the Lyapunov structure and achieving unconditional long-time stability. It proves that TEM and Galerkin-based discretizations inherit geometric ergodicity under nondegeneracy and derives strong error bounds in the multiplicative-noise setting, with optimal convergence rates for additive noise. The results cover stochastic Allen–Cahn dynamics in dimensions up to three and provide high-order convergence guarantees under strengthened regularity, offering practical, nonlinearity-explicit integrators for ergodic SPDEs. The method delivers efficient, provably accurate long-time simulations and invariant-measure approximations for SPDEs with non-Lipschitz nonlinearities.

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.

Paper Structure

This paper contains 13 sections, 15 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Assumptions
  5. Tamed Schemes
  6. Geometric Ergodicity of Tamed Schemes
  7. Lyapunov Structure
  8. Application to Galerkin-based tamed full discretizations
  9. Geometric Ergodicity of GTEM and Drift-GTEM
  10. Strong Error Estimates in Multiplicative Noise Case
  11. Auxiliary Process and Moment's Estimates
  12. Strong Error Estimate
  13. High-order Convergence Rate in Additive Noise Case

Key Result

Lemma 3.1

Under the conditions coe-f-grow, for all $\beta \in (0, L_2)$, $\tau \in (0, (L_2-\beta)^2 L_4^{-4}/8]$, and $\alpha>0$, there exist positive constants ${\widetilde{L}}_3$ and ${\widetilde{L}}_4$ depending on $L_3$ and $L_4$ such that for any $u \in \dot H^1$,

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • ...and 26 more