Numerical Method for Space-Time Fractional Diffusion: A Stochastic Approach

Abstract: In this paper, we develop and analyze a stochastic algorithm for solving space-time fractional diffusion models, which are widely used to describe anomalous diffusion dynamics. These models pose substantial numerical challenges due to the memory effect of the time-fractional derivative and the nonlocal nature of the spatial fractional Laplacian and the, leading to significant computational costs and storage demands, particularly in high-dimensional settings. To overcome these difficulties, we propose a Monte Carlo method based on the Feynman--Kac formula for space-time fractional models. The novel algorithm combines the simulation of the monotone path of a stable subordinator in time with the ``walk-on-spheres'' method that efficiently simulates the stable Levy jumping process in space. We rigorously derive error bounds for the proposed scheme, explicitly expressed in terms of the number of simulation paths and the time step size. Numerical experiments confirm the theoretical error bounds and demonstrate the computational efficiency of the method, particularly in domains with complex geometries or high-dimensional spaces. Furthermore, both theoretical and numerical results emphasize the robustness of the proposed approach across a range of fractional orders, particularly for small fractional values, a capability often absent in traditional numerical methods.