A high accuracy method for the nonlinear fractional diffusion problem on network

Yangming Zhang; Yan Fan; Yangming Zhang; Yan Fan

doi:10.3934/era.2025276

Electronic Research Archive

2025, Volume 33, Issue 10: 6241-6266. doi: 10.3934/era.2025276

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A high accuracy method for the nonlinear fractional diffusion problem on network

Yangming Zhang ¹,
Yan Fan ^{2
,
,}

1.
School of Mathematics and Statistics, Hunan First Normal University, Changsha 410006, China
2.
Xingzhi College, Zhejiang Normal University, Jinhua 321004, China

Received: 17 September 2025 Revised: 13 October 2025 Accepted: 16 October 2025 Published: 22 October 2025

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 $ \alpha $. Finally, several numerical experiments are presented to verify and support the theoretical results.
- network dynamics,
- fractional random walk,
- fractional diffusion,
- high accuracy numerical method,
- convergence and stability
Citation: Yangming Zhang, Yan Fan. A high accuracy method for the nonlinear fractional diffusion problem on network[J]. Electronic Research Archive, 2025, 33(10): 6241-6266. doi: 10.3934/era.2025276

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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 $ \alpha $. Finally, several numerical experiments are presented to verify and support the theoretical results.

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