Robust numerical framework for simulating 2D fractional time–space stochastic diffusion equation driven by spatio-temporal noise: L1-FFT hybrid approach

Abstract

This paper presents an in-depth analysis of numerical methods for solving two-dimensional fractional time–space stochastic diffusion equations, employing the Caputo fractional derivative and the fractional Laplacian. The study utilizes the L1-algorithm for temporal discretization, ensuring an accurate representation of fractional dynamics, while the Fast Fourier Transform is applied for spatial discretization to efficiently handle large-scale computational challenges. The spatio-temporal multiplicative Gaussian noise is modeled as the product of a Brownian sheet and a Brownian bridge, with the 2D Karhunen–Loève expansion implemented for noise generation. This approach is evaluated against Euler-type simulation technique, establishing a robust framework for stochastic simulations. Through comprehensive testing using two benchmark initial conditions — the sinc function and a single Gaussian distribution — the effectiveness of the proposed approach was validated in resolving two-dimensional fractional stochastic diffusion equations and simulating pollutant concentration dispersion patterns. These advanced methodologies enhance the understanding of fractional diffusion processes and offer practical tools for addressing complex problems in environmental modeling and related fields.