Long time stability and strong convergence of an efficient tamed scheme for stochastic Allen-Cahn equation driven by additive white noise

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2026-06-01 Epub Date: 2026-01-27 DOI:10.1016/j.apnum.2026.01.017

Xiao Qi , Yubin Yan

{"title":"Long time stability and strong convergence of an efficient tamed scheme for stochastic Allen-Cahn equation driven by additive white noise","authors":"Xiao Qi , Yubin Yan","doi":"10.1016/j.apnum.2026.01.017","DOIUrl":null,"url":null,"abstract":"<div><div>Huang and Shen [<em>Math. Comput.</em> <strong>92</strong> (2023) 2685–2713] proposed a semi-implicit tamed scheme for the numerical approximation of stochastic Allen–Cahn equations driven by multiplicative trace-class noise. They showed that the scheme is unconditionally stable on finite time intervals and can be efficiently implemented. In this paper, we investigate the long-time stability of this tamed scheme for stochastic Allen–Cahn equations driven by additive white noise. We also address the strong convergence analysis of the associated fully discrete scheme within the Galerkin finite element framework. The main contributions of this work are as follows: (i) by constructing a suitable Lyapunov functional, we establish the unconditional long-time stability of the tamed method; (ii) we rigorously derive the strong convergence rates of the fully discrete scheme obtained by coupling the tamed approach with the finite element method. Numerical experiments are provided to validate the theoretical analysis and demonstrate the effectiveness of the proposed scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"224 ","pages":"Pages 22-36"},"PeriodicalIF":2.4000,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927426000243","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/1/27 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

Huang and Shen [Math. Comput. 92 (2023) 2685–2713] proposed a semi-implicit tamed scheme for the numerical approximation of stochastic Allen–Cahn equations driven by multiplicative trace-class noise. They showed that the scheme is unconditionally stable on finite time intervals and can be efficiently implemented. In this paper, we investigate the long-time stability of this tamed scheme for stochastic Allen–Cahn equations driven by additive white noise. We also address the strong convergence analysis of the associated fully discrete scheme within the Galerkin finite element framework. The main contributions of this work are as follows: (i) by constructing a suitable Lyapunov functional, we establish the unconditional long-time stability of the tamed method; (ii) we rigorously derive the strong convergence rates of the fully discrete scheme obtained by coupling the tamed approach with the finite element method. Numerical experiments are provided to validate the theoretical analysis and demonstrate the effectiveness of the proposed scheme.

查看原文本刊更多论文

加性白噪声驱动下随机Allen-Cahn方程的一种有效驯服格式的长时间稳定性和强收敛性

黄和沈[数学]。[j] .计算机学报，92(2023)2685-2713]提出了一种由乘性迹类噪声驱动的随机Allen-Cahn方程数值逼近的半隐式拟合格式。结果表明，该方案在有限时间内是无条件稳定的，可以有效地实现。本文研究了加性白噪声驱动下随机Allen-Cahn方程的这种驯服格式的长期稳定性。本文还讨论了在Galerkin有限元框架下相关的全离散格式的强收敛性分析。本工作的主要贡献如下：(1)通过构造合适的Lyapunov泛函，我们建立了驯服方法的无条件长期稳定性；（ii）严格推导了由驯服方法与有限元方法耦合得到的完全离散格式的强收敛率。数值实验验证了理论分析和所提方案的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.