Unconditionally stable and high-order numerical method for variable-order Caputo-Fabrizio fractional diffusion equations

Xiao Chen; Shanshan Wang; Lingna Lu; Xiao Chen; Shanshan Wang; Lingna Lu

doi:10.3934/nhm.2026078

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2068-2083. doi: 10.3934/nhm.2026078

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Unconditionally stable and high-order numerical method for variable-order Caputo-Fabrizio fractional diffusion equations

1.
School of General Education, Zhengzhou Business University, Zhengzhou 450046, China
2.
School of Statistics, Henan University of Economics and Law, Zhengzhou 450046, China
3.
School of Basic Sciences, Zhengzhou University of Technology, Zhengzhou 450044, China

Received: 15 May 2026 Revised: 08 July 2026 Accepted: 21 July 2026 Published: 20 October 2026

In this work, we present and analyze a high-precision numerical scheme to solve a class of variable-order fractional diffusion equations involving the Caputo-Fabrizio (CF) fractional derivative. The proposed scheme employs a finite difference method for temporal discretization and a local discontinuous Galerkin (LDG) method equipped with generalized alternating numerical fluxes for spatial discretization. For any variable-order function $ \alpha(t) \in (0, 1) $, we demonstrate the stability and $ L^2 $ convergence of the proposed scheme. Moreover, we show that the method attains an accuracy order of $ O(\tau^2 + h^{k+1}) $, where $ \tau $, $ h $, and $ k $ denote the temporal step size, spatial step size, and degree of the piecewise $ P^k $ polynomials, respectively. A series of numerical experiments are carried out to verify the theoretical findings and the effectiveness of the proposed algorithm.
- variable-order derivatives,
- finite difference method,
- local discontinuous Galerkin method,
- stability,
- error estimates
Citation: Xiao Chen, Shanshan Wang, Lingna Lu. Unconditionally stable and high-order numerical method for variable-order Caputo-Fabrizio fractional diffusion equations[J]. Networks and Heterogeneous Media, 2026, 21(6): 2068-2083. doi: 10.3934/nhm.2026078

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Abstract

In this work, we present and analyze a high-precision numerical scheme to solve a class of variable-order fractional diffusion equations involving the Caputo-Fabrizio (CF) fractional derivative. The proposed scheme employs a finite difference method for temporal discretization and a local discontinuous Galerkin (LDG) method equipped with generalized alternating numerical fluxes for spatial discretization. For any variable-order function $ \alpha(t) \in (0, 1) $, we demonstrate the stability and $ L^2 $ convergence of the proposed scheme. Moreover, we show that the method attains an accuracy order of $ O(\tau^2 + h^{k+1}) $, where $ \tau $, $ h $, and $ k $ denote the temporal step size, spatial step size, and degree of the piecewise $ P^k $ polynomials, respectively. A series of numerical experiments are carried out to verify the theoretical findings and the effectiveness of the proposed algorithm.

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