A structure-preserving reduced-order finite difference approach for a class of semilinear stochastic partial differential equations driven by white noise

This paper presents a novel reduced-order finite difference (ROFD) approach that integrates proper orthogonal decomposition (POD) with the finite difference (FD) method to efficiently solve a class of semilinear stochastic partial differential equations (SPDEs) driven by white noise. SPDEs are a class of mathematical models that incorporate random terms or stochastic processes to describe the evolution of systems under uncertainty. SPDEs play a crucial role in modeling real-world phenomena across various fields, including physics, finance, and environmental science, where stochastic process is an inherent component. However, the presence of noise terms and selection of large sample data pose significant challenges for numerical solutions. The proposed ROFD method not only retains the approximation accuracy of the original FD method but also preserves the structural properties of the original semilinear SPDEs. For instance, the mathematical expectation of the numerical solutions under large-sample data satisfies the maximum principle, energy dissipation and so on. A series of numerical experiments have been performed to evaluate the effectiveness of the ROFD method in solving a class of semilinear SPDEs. The numerical results demonstrate that the ROFD method provides highly accurate numerical solutions, exhibits excellent stability and significantly enhances computational efficiency. Due to these advantages, it serves as a highly competitive and practical numerical method for addressing complex SPDEs in real-world applications.