Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 8 2 2024 09 25 Upwind Implicit Scheme for the Numerical Solution of Stochastic Advection-Diffusion Partial Differential Equations 138 162 434 10.22128/gadm.2024.859.1118 EN Mehran Namjoo Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran 0000-0001-5949-6766 Mehran Aminian Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Ali Mohebbian Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Mehdi Karami Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Hossein Salmei Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Journal Article 2024 08 10 Stochastic partial differential equations (SPDEs) are significant in various fields such as epidemiology‎, ‎mechanics‎, ‎microelectronics‎, ‎chemistry‎, ‎and finance‎. ‎Obtaining analytical solutions for SPDEs is either difficult or impossible; therefore‎, ‎researchers are very interested in effective numerical methods for studying the behavior of these equations‎. ‎In this paper‎, ‎we introduce a stochastic finite difference (SFD) scheme for the numerical solution of the It\^{o} stochastic advection--diffusion equation‎. ‎We discuss the consistency‎, ‎stability‎, ‎and convergence of the scheme‎, ‎and we also determine its order of convergence‎. ‎Finally‎, ‎to validate the effectiveness and accuracy of the SFD scheme‎, ‎we analyze the numerical results and compare them with those from existing SFD schemes. https://ansne.du.ac.ir/article_434_db2592f41b970d7c3f1ff0afd0dd4bf2.pdf