Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α\alpha (1<α<21\lt \alpha \lt 2). By combining the compact difference approach for spatial discretization and the alternating dir...
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De Gruyter
2021
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oai:doaj.org-article:2377214dcc3a4119be6a880a4dccf0902021-12-05T14:10:53ZNumerical methods for time-fractional convection-diffusion problems with high-order accuracy2391-545510.1515/math-2021-0036https://doaj.org/article/2377214dcc3a4119be6a880a4dccf0902021-08-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0036https://doaj.org/toc/2391-5455In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α\alpha (1<α<21\lt \alpha \lt 2). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H1{H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O(τ3−α+h14+h24)O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}), where τ\tau is the temporal grid size and h1{h}_{1}, h2{h}_{2} are spatial grid sizes in the xx and yy directions, respectively. It is proved that the method can even attain (1+α)\left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.Dong GangGuo ZhichangYao WenjuanDe Gruyterarticle2d time-fractional convection-diffusion equationadi schemeunconditional stabilityconvergence37mxx65-xxMathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 782-802 (2021) |
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EN |
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2d time-fractional convection-diffusion equation adi scheme unconditional stability convergence 37mxx 65-xx Mathematics QA1-939 |
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2d time-fractional convection-diffusion equation adi scheme unconditional stability convergence 37mxx 65-xx Mathematics QA1-939 Dong Gang Guo Zhichang Yao Wenjuan Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| description |
In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α\alpha (1<α<21\lt \alpha \lt 2). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H1{H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O(τ3−α+h14+h24)O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}), where τ\tau is the temporal grid size and h1{h}_{1}, h2{h}_{2} are spatial grid sizes in the xx and yy directions, respectively. It is proved that the method can even attain (1+α)\left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis. |
| format |
article |
| author |
Dong Gang Guo Zhichang Yao Wenjuan |
| author_facet |
Dong Gang Guo Zhichang Yao Wenjuan |
| author_sort |
Dong Gang |
| title |
Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| title_short |
Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| title_full |
Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| title_fullStr |
Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| title_full_unstemmed |
Numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| title_sort |
numerical methods for time-fractional convection-diffusion problems with high-order accuracy |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/2377214dcc3a4119be6a880a4dccf090 |
| work_keys_str_mv |
AT donggang numericalmethodsfortimefractionalconvectiondiffusionproblemswithhighorderaccuracy AT guozhichang numericalmethodsfortimefractionalconvectiondiffusionproblemswithhighorderaccuracy AT yaowenjuan numericalmethodsfortimefractionalconvectiondiffusionproblemswithhighorderaccuracy |
| _version_ |
1718371589930614784 |