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Uncoupling evolutionary groundwater-surface water flows: stabilized mixed methods in both porous media and fluid regions

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Abstract

This paper considers the robust numerical methods for solving the time-dependent Stokes-Darcy multiphysics problem that can be implemented by use of existing surface water and groundwater codes. Porous media problem for the groundwater flow is preferable to employ the mixed discretization due to their superior conservation property and the convenience to compute flux on the large domain with relatively coarse meshes. However, the theory of mixed spatial discretizations for the time-dependent problems is far less developed than the non-mixed approaches, even for the one domain problems. Herein, we develop a stabilized mixed discretization technique for the porous media problem coupled with the fluid region across an interface with the physically appropriate coupling conditions. Time discretization is constructed to allow a non-iterative splitting of the coupled problem into two subproblems. The stability and convergence analysis of the coupled and decoupled algorithms are derived rigorously. If the time scale is bounded by a constant which only depends on the physical parameters, we prove the unconditional stability of both schemes. Four numerical experiments are conducted to show the efficiency and accuracy of the numerical methods, which illustrate the exclusive features of the Stokes-Darcy interface system.

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Funding

H. Zheng is partially supported by NSF of China (Grant No. 11971174), NSF of Shanghai (Grant No. 19ZR1414300) and Science and Technology Commission of Shanghai Municipality (Grant Nos. 22JC1400900, 21JC1402500, 18dz2271000). L. Shan is supported by Shantou University Scientific Research Fund for Talents (NTF21006) and Scientific Research Fund of Liaoning Provincial Education Department (LJ2020JCL009).

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Authors and Affiliations

  1. Department of Mathematics, Comilla University, Cumilla, 3506, Bangladesh

    Md. Abdullah Al Mahbub

  2. Department of Mathematics, Shantou University, Shantou, 515063, China

    Li Shan

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, People’s Republic of China

    Haibiao Zheng

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  1. Md. Abdullah Al Mahbub
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  2. Li Shan
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  3. Haibiao Zheng
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Correspondence to Li Shan.

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Mahbub, M.A.A., Shan, L. & Zheng, H. Uncoupling evolutionary groundwater-surface water flows: stabilized mixed methods in both porous media and fluid regions. Numer Algor 92, 1837–1874 (2023). https://doi.org/10.1007/s11075-022-01370-3

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