Optimal Strong Convergence Rate of Spectral Galerkin Exponential Euler Scheme for Parabolic SPDEs

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Abstract

We provide a new approach to strong error analysis of the spatial-spectral Galerkin and temporal exponential Euler scheme for a family of second-order parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise. Applying these results to the stochastic advection-diffusion-reaction equation with a gradient term driven by white noise indicates that this scheme achieves optimal strong convergence order exactly 1/2 in space, which removes an infinitesimal factor in the literature, and 1/4 in time. Numerical experiments support our theoretical analysis.

Keywords:

Parabolic stochastic partial differential equation multiplicative white noise spectral Galerkin approximation exponential integrator strong convergence rate

Author Biographies

  • Jialin Hong

    LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China
    School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.R. China

  • Chuying Huang

    School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Key Laboratory of Analytical Mathematics and Applications, Center for Applied Mathematics of Fujian Province, Fujian Normal University, Fuzhou 350117, P.R. China.

  • Zhihui Liu

    Department of Mathematics & National Center for Applied Mathematics Shenzhen (NCAMS) & Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R. China.

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DOI

10.4208/csiam-am.SO-2025-0040

How to Cite

Optimal Strong Convergence Rate of Spectral Galerkin Exponential Euler Scheme for Parabolic SPDEs. (2026). CSIAM Transactions on Applied Mathematics. https://doi.org/10.4208/csiam-am.SO-2025-0040