<p>In this paper, we study numerical methods for the stochastic Allen–Cahn equation driven by multiplicative trace-class noise. The temporal discretization uses a drift-implicit Euler scheme, and the spatial discretization employs a spectral Galerkin method. We show that the spatial weak convergence rate is nearly one order higher than the corresponding strong convergence rate for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and nearly <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation> order higher than the corresponding strong convergence rate for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>; and that the temporal weak convergence rate is close to order one for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and close to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{3}{4}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. The weak error analysis is carried out by deriving a priori estimates for the solutions of the Kolmogorov equations associated with the spectral Galerkin semi-discretization. We also develop techniques to handle the trace of an operator involving a stochastic integral for the temporal weak error analysis. Finally, numerical experiments are presented to illustrate the theoretical results.</p>

Weak Convergence of Drift-Implicit Euler and Spectral Galerkin Approximation to Stochastic Allen–Cahn Equation Driven by Multiplicative Trace-Class Noise

期刊论文
出版模式：

开放获取
发表日期：

2026年3月18日

查看全文

Minxing Zhang,
Yongkui Zou,
Ran Zhang,
Yanzhao Cao

摘要

In this paper, we study numerical methods for the stochastic Allen–Cahn equation driven by multiplicative trace-class noise. The temporal discretization uses a drift-implicit Euler scheme, and the spatial discretization employs a spectral Galerkin method. We show that the spatial weak convergence rate is nearly one order higher than the corresponding strong convergence rate for $d=1,2$ $d = 1, 2$ , and nearly $\frac{1}{2}$ $\frac{1}{2}$ order higher than the corresponding strong convergence rate for $d=3$ $d = 3$ ; and that the temporal weak convergence rate is close to order one for $d=1,2$ $d = 1, 2$ and close to $\frac{3}{4}$ $\frac{3}{4}$ for $d=3$ $d = 3$ . The weak error analysis is carried out by deriving a priori estimates for the solutions of the Kolmogorov equations associated with the spectral Galerkin semi-discretization. We also develop techniques to handle the trace of an operator involving a stochastic integral for the temporal weak error analysis. Finally, numerical experiments are presented to illustrate the theoretical results.