An accurate parallel finite difference method for the variable-order fractional Laplacian in heterogeneous media

Abstract

Anomalous diffusion in heterogeneous media is ubiquitous across physical and biological systems, where spatially varying microstructures give rise to distinct transport regimes. Variable-order nonlocal operators provide a natural modeling framework for describing heterogeneity and transitions between diffusion regimes. However, the position-dependent long-range interactions inherent in these operators impose prohibitive memory costs and make numerical methods developed for constant-order operators computationally impractical. Consequently, accurate and computationally efficient numerical methods for variable-order nonlocal operators remain scarce. In this paper, we propose accurate finite difference discretizations for the variable-order fractional Laplacian $(-Δ)^{α(\mathbf{x})/2}$ and develop a fast Fourier transform (FFT)-accelerated scheme for their efficient implementation. The computational and memory costs are $O\!\big(MN\log(2N)\big)$ and $O(MN)$, respectively, where $N$ is the total number of spatial points and $M \ll N$. The method is also naturally parallelizable. We establish its consistency and accuracy through a rigorous error analysis that quantifies the local interplay between solution regularity and the spatially varying order $α(\mathbf{x})$. To the best of our knowledge, this is the first work to quantify how these two factors jointly determine convergence rates, with numerical experiments demonstrating this dependence. Extensive numerical experiments confirm the accuracy and efficiency of the proposed method. Applications to anomalous diffusion in composite materials further demonstrate its ability to capture the effects of material heterogeneity.