Upwind Implicit Scheme for the Numerical Solution of Stochastic Advection–Diffusion Partial Differential Equations

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.