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Compact finite-difference method for 2D time-fractional convection–diffusion equation of groundwater pollution problems

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Abstract

In this work, we provide a compact finite-difference scheme (CFDS) of 2D time-fractional convection–diffusion equation (TF-CDE) for solving fluid dynamics problem, especially groundwater pollution. The successful predication of the pollutants concentration in groundwater will greatly benefit the protection of water resources for provide the fast and intuitive decision-makings in response to sudden water pollution events. Here, we creatively use the dimensionality reduction technology (DRT) to rewrite the original 2D problem as two equations, and we handle each one as a 1D problem. Particularly, the spatial derivative is approximated by fourth-order compact finite-difference method (CFDM) and time-fractional derivative is approximated by \(L_{1}\) interpolation of Caputo fractional derivative. Based on the approximations, we obtain the CFDS with fourth-order in spatial and \((2-\alpha )\)-order in temporal by adding two 1D results. In addition, the unique solvability, unconditional stability, and convergence order \({\mathscr {O}}(\tau ^{2-\alpha } + h_{1}^4 + h_{2}^4)\) of the proposed scheme are studied. Finally, several numerical examples are carried out to support the theoretical results and demonstrate the effectiveness of the CFDS based DRT strategy. Obviously, the method developed in 2D TF-CDE of groundwater pollution problem can be easily extended for the other complex problems.

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Acknowledgements

The authors are very grateful to Dr. Haili Qiao of Shandong University for her assistance in the process of numerical calculation. We are also very grateful to the anonymous reviewers for their invaluable time and insightful comments leading to the improved manuscript.

Funding

This work was partially supported by the National Natural Science Foundation of China (NSFC) under Grant Numbers 11501335 and 11371229, and the Natural Science Foundation of Shandong Province, China under Grant Numbers ZR2017MA020, ZR2017MA003.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Lingyu Li, Ziwen Jiang & Zhe Yin

  2. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Lingyu Li

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  1. Lingyu Li
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  2. Ziwen Jiang
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Correspondence to Ziwen Jiang or Zhe Yin.

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Communicated by José Tenreiro Machado.

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Li, L., Jiang, Z. & Yin, Z. Compact finite-difference method for 2D time-fractional convection–diffusion equation of groundwater pollution problems. Comp. Appl. Math. 39, 142 (2020). https://doi.org/10.1007/s40314-020-01169-9

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  • DOI: https://doi.org/10.1007/s40314-020-01169-9

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