Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-02-18 DOI:10.1137/24m1636691

Xiaochen Yang, Yaozhong Hu

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Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 396-421, February 2025.
Abstract. This paper studies the long time stability of both the solution of a stochastic heat equation on a bounded domain driven by a correlated noise and its approximations. It is popular for researchers to prove the intermittency of the solution, which means that the moments of solution to a stochastic heat equation usually grow to infinity exponentially fast and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation, which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on a bounded domain to write a stochastic heat equation as a system of infinite many stochastic differential equations. We also present numerical experiments which are consistent with our theoretical results.

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随机热方程隐式格式的长时间稳定性和数值稳定性

SIAM数值分析杂志，第63卷，第1期，第396-421页，2025年2月。摘要。本文研究了由相关噪声驱动的有界区域上随机热方程解及其近似解的长时间稳定性。研究人员普遍认为，解的间歇性是一个普遍的问题，这意味着随机热方程的解的矩通常以指数速度增长到无穷大，这暗示了随机热方程的解在长时间内通常是不稳定的。然而，令人惊讶的是，在本文中，我们证明了当域是有界的，当噪声在空间变量中不是奇异时，系统可以长时间稳定，并且我们还证明了我们可以用它的有限维谱近似来近似解，这也是长时间稳定的。其思想是利用拉普拉斯函数在有界域上的特征函数展开，将一个随机热方程写成一个由无穷多个随机微分方程组成的系统。我们还提出了与理论结果一致的数值实验。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.