@article{Zhang2025, 
author = {Yangming Zhang and Yan Fan},
title = {A high accuracy method for the nonlinear fractional diffusion problem on network},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {10},
pages = {6241-6266},
keywords = {network dynamics, fractional random walk, fractional diffusion, high accuracy numerical method, convergence and stability},
url = {https://www.sciopen.com/article/10.3934/era.2025276},
doi = {10.3934/era.2025276},
abstract = {The fractional diffusion equation involving the fractional Laplacian is used to govern fractional random walk dynamics on network, which allowing long-range displacements. This paper develops a high accuracy numerical method for the computation of nonlinear fractional diffusion equation. The main idea is to approximate the spatial domain with a spectral Galerkin method based on Fourier-like basis functions, and then to discretize time by the general linear methods which contains Runge-Kutta methods, multistep methods, and many new classes of methods. For    (  k  ,  l  )-algebraically stable general linear methods with general stage order    p, the nonlinear term satisfies the locally Lipschitz condition, and the proposed method is proved to be well-posed, stable, and convergent with order    p in time. Moreover, an optimal spatial error estimate is established, whose convergence rate is independent of the fractional parameter    α. Finally, several numerical experiments are presented to verify and support the theoretical results.}
}