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Strong convergence rates for full-discrete approximations of the stochastic Allen-Cahn equations on 2D torus

Ting Ma, Lifei Wang, Huanyu Yang

TL;DR

This work addresses numerical approximation of the 2D stochastic Allen–Cahn equation on the torus driven by space–time white noise, a singular SPDE that requires Wick renormalization. The authors develop a space–time full discretization combining a spectral Galerkin method for the linear component and a taming-based exponential Euler scheme for the nonlinear part, with integration starting at a positive time to deal with initial singularities. By leveraging Besov-space techniques and a Da Prato–Debussche decomposition $X=Y+\\bar{Z}$, they prove strong convergence with spatial rate $α-\\delta$ and temporal rate $α/6-\\delta$ in $\\mathcal{C}^{-\\alpha}$ for $\\alpha \\in (0,1/3)$ and arbitrary small $\\delta>0$, and provide corresponding a priori bounds for the nonlinear component. The results advance numerical analysis for high-dimensional SPDEs with superlinear nonlinearities and rough noise, offering rigorous guarantees for practical space–time discretizations. The methodology and error estimates inform robust simulations of singular SPDEs in two dimensions and beyond.

Abstract

In this paper we construct space-time full discretizations of stochastic Allen-Cahn equations driven by space-time white noise on 2D torus. The approximations are implemented by tamed exponential Euler discretization in time and spectral Galerkin method in space. We finally obtain the convergence rates with the spatial order of $α-δ$ and the temporal order of $α/{6}-δ$ in $\mathcal C^{-α}$ for $α\in(0,1/3)$ and $δ>0$ arbitrarily small.

Strong convergence rates for full-discrete approximations of the stochastic Allen-Cahn equations on 2D torus

TL;DR

This work addresses numerical approximation of the 2D stochastic Allen–Cahn equation on the torus driven by space–time white noise, a singular SPDE that requires Wick renormalization. The authors develop a space–time full discretization combining a spectral Galerkin method for the linear component and a taming-based exponential Euler scheme for the nonlinear part, with integration starting at a positive time to deal with initial singularities. By leveraging Besov-space techniques and a Da Prato–Debussche decomposition , they prove strong convergence with spatial rate and temporal rate in for and arbitrary small , and provide corresponding a priori bounds for the nonlinear component. The results advance numerical analysis for high-dimensional SPDEs with superlinear nonlinearities and rough noise, offering rigorous guarantees for practical space–time discretizations. The methodology and error estimates inform robust simulations of singular SPDEs in two dimensions and beyond.

Abstract

In this paper we construct space-time full discretizations of stochastic Allen-Cahn equations driven by space-time white noise on 2D torus. The approximations are implemented by tamed exponential Euler discretization in time and spectral Galerkin method in space. We finally obtain the convergence rates with the spatial order of and the temporal order of in for and arbitrarily small.
Paper Structure (14 sections, 19 theorems, 151 equations)

This paper contains 14 sections, 19 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main methods and results
  4. Structure of the paper
  5. Besov spaces and preliminaries
  6. Stochastic heat equation
  7. Wick powers and Galerkin approximation
  8. Regularity results
  9. Proof of main result
  10. Space-time full discretization
  11. A priori bounds for the approximations
  12. Strong convergence rates
  13. Space-time white noise and Wiener chaos
  14. Functions with prescribed singularities

Key Result

Theorem 1.1

Let $X_0\in\mathcal{C}^{-\alpha}$ with $\alpha\in (0,1/3)$, $\gamma>1-3\alpha$ and $p\geqslant2$. Then for any $\delta>0$ for uniform large $N,M\in \mathbb N$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 19 more