Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data

作     者：Xinping Gui Buyang Li Jilu Wang 

作者机构：Beijing Computational Science Research Center Department of Applied Mathematics The Hong Kong Polytechnic University School of Science Harbin Institute of Technology 

出 版 物：《Science China Mathematics》 (中国科学:数学(英文版))

年 卷 期：2024年

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 12071020, 12131005 and U2230402) the Research Grants Council of Hong Kong (Grant No. PolyU15300519) an Internal Grant of The Hong Kong Polytechnic University (Grant No. P0038843, Work Programme: ZVX7) 

摘      要：A class of stochastic Besov spaces BpL2(Ω;H~α(O)), 1≤ p ≤∞ and α ∈ [-2, 2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du-△udt = f(u)dt + d W(t),under the following conditions for some α ∈(0, 1]:‖(∫t)0e~(-(t-s)A)dW(S)‖_(L2(Ω;L2(O)))≤Cta/2and ‖(∫t)0e~(-(t-s)A)dW(S)‖_(B~∞L2(Ω;H~α(O)))≤C The conditions above are shown to be satisfied by both trace-class noises(with α = 1) and one-dimensional space-time white noises(with α =1/2). The latter would fail to satisfy the conditions with α =1/2 if the stochastic Besov norm ∥·∥_(B~∞L2(Ω;H~α(O)))is replaced by the classical Sobolev norm ∥·∥_(L2(Ω;H~α(O))), and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this paper, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization,is proved to have order α in both the time and space for possibly nonsmooth initial data in L4(Ω;H~β(O)) withβ -1, by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t = 0.