A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative

Lijie Liu; Xiaojing Wei; Leilei Wei; Lijie Liu; Xiaojing Wei; Leilei Wei

doi:10.3934/era.2023289

Electronic Research Archive

2023, Volume 31, Issue 9: 5701-5715. doi: 10.3934/era.2023289

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Research article

A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative

College of Science, Henan University of Technology, Zhengzhou 450001, P.R. China

Received: 26 June 2023 Revised: 27 July 2023 Accepted: 03 August 2023 Published: 17 August 2023

In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For $ 0 < \alpha(t) < 1 $, we prove that the method is unconditionally stable and the errors attain $ (k+1) $-th order of accuracy for piecewise $ P^k $ polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.
- Caputo fractional derivative,
- generalized alternating numerical flux,
- stability,
- error estimate
Citation: Lijie Liu, Xiaojing Wei, Leilei Wei. A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative[J]. Electronic Research Archive, 2023, 31(9): 5701-5715. doi: 10.3934/era.2023289

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Abstract

In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For $ 0 < \alpha(t) < 1 $, we prove that the method is unconditionally stable and the errors attain $ (k+1) $-th order of accuracy for piecewise $ P^k $ polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.

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