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
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 f...
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AIMS Press
2023-08-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023289?viewType=HTML |
| _version_ | 1797657982955159552 |
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| author | Lijie Liu Xiaojing Wei Leilei Wei |
| author_facet | Lijie Liu Xiaojing Wei Leilei Wei |
| author_sort | Lijie Liu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T17:52:34Z |
| format | Article |
| id | doaj.art-e52cce04a02e41b0a8774bec1cfb36bf |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-11T17:52:34Z |
| publishDate | 2023-08-01 |
| publisher | AIMS Press |
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| spelling | doaj.art-e52cce04a02e41b0a8774bec1cfb36bf2023-10-18T01:56:44ZengAIMS PressElectronic Research Archive2688-15942023-08-013195701571510.3934/era.2023289A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivativeLijie Liu0Xiaojing Wei1Leilei Wei2College of Science, Henan University of Technology, Zhengzhou 450001, P.R. ChinaCollege of Science, Henan University of Technology, Zhengzhou 450001, P.R. ChinaCollege of Science, Henan University of Technology, Zhengzhou 450001, P.R. ChinaIn 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.https://www.aimspress.com/article/doi/10.3934/era.2023289?viewType=HTMLcaputo fractional derivativegeneralized alternating numerical fluxstabilityerror estimate |
| spellingShingle | 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 Electronic Research Archive caputo fractional derivative generalized alternating numerical flux stability error estimate |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | fully discrete local discontinuous galerkin method based on generalized numerical fluxes to variable order time fractional reaction diffusion problem with the caputo fractional derivative |
| topic | caputo fractional derivative generalized alternating numerical flux stability error estimate |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023289?viewType=HTML |
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