Numerical approximation of fourth-order fractional diffusion-wave systems using finite difference and discontinuous Galerkin method

Chuan Ran; Xiaoyan Xu; Changshun Hou; Xindong Zhang; Chuan Ran; Xiaoyan Xu; Changshun Hou; Xindong Zhang

doi:10.3934/nhm.2025058

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1346-1366. doi: 10.3934/nhm.2025058

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Numerical approximation of fourth-order fractional diffusion-wave systems using finite difference and discontinuous Galerkin method

1.
College of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
School of Mathematics and Statistics, Henan University of Technology, Zhengzhou 450001, China

Received: 08 September 2025 Revised: 27 November 2025 Accepted: 04 December 2025 Published: 10 December 2025
65M12; 65M06; 35S10

This study develops a finite difference/local discontinuous Galerkin (LDG) framework for solving a fourth-order fractional diffusion-wave equation. The temporal fractional operator is approximated through a finite difference approach, achieving a truncation accuracy of $ O((\Delta t)^{3-\alpha}) $, where $ \Delta t $ denotes the time increment and $ \alpha $ represents the fractional order. For spatial discretization, LDG technique is employed, which leads to a fully implicit discrete formulation of the considered model. By applying mathematical induction, we establish the unconditional stability and convergence of the proposed algorithm. A series of computational experiments is presented to verify the theoretical error bounds.
- fourth-order fractional diffusion-wave system,
- time fractional derivative,
- stability,
- convergence
Citation: Chuan Ran, Xiaoyan Xu, Changshun Hou, Xindong Zhang. Numerical approximation of fourth-order fractional diffusion-wave systems using finite difference and discontinuous Galerkin method[J]. Networks and Heterogeneous Media, 2025, 20(4): 1346-1366. doi: 10.3934/nhm.2025058

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Abstract

This study develops a finite difference/local discontinuous Galerkin (LDG) framework for solving a fourth-order fractional diffusion-wave equation. The temporal fractional operator is approximated through a finite difference approach, achieving a truncation accuracy of $ O((\Delta t)^{3-\alpha}) $, where $ \Delta t $ denotes the time increment and $ \alpha $ represents the fractional order. For spatial discretization, LDG technique is employed, which leads to a fully implicit discrete formulation of the considered model. By applying mathematical induction, we establish the unconditional stability and convergence of the proposed algorithm. A series of computational experiments is presented to verify the theoretical error bounds.

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