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Stabilized Variational Formulations of Chorin-Type and Artificial Compressibility Methods for the Stochastic Stokes–Darcy Equations

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Abstract

In this paper, we consider two different types of numerical schemes for the nonstationary stochastic Stokes–Darcy equations with multiplicative noise. Firstly, we consider the Chorin-type time-splitting scheme for the Stokes equation in the free fluid region. The Darcy equation and an elliptic equation for the intermediate velocity of free fluid coupled with the interface conditions are solved, and then the velocity and pressure in free fluid region are updated by an elliptic system. Secondly, we further consider the artificial compressibility method (ACM) which separates the fully coupled Stokes–Darcy model into two smaller subphysics problems. The ACM reduces the storage and the computational time at each time step, and allows parallel computing for the decoupled problems. The pressure in free fluid region only needs to be updated at each time step without solving an elliptic system. We utilize the RT\(_1\)-P\(_1\) pair finite element space and the interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM\(_1\)-P\(_0\) finite element space in the spatial discretizations. Under usual assumptions for the multiplicative noise, we prove that both of the Chorin-type scheme and the ACM are unconditionally stable. We present the error estimates for the time semi-discretization of the Chorin-type scheme. Numerical examples are provided to verify the stability estimates for both of schemes. Moreover, we test the convergence rate for the velocity in time for both of schemes, and the convergence rate for the pressure approximation in time average is also tested.

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Funding

The work of Huangxin Chen was supported by the National Key Research and Development Project of China (Grant No. 2023YFA1011702) and the National Natural Science Foundation of China (Grant No. 12122115). The work of Can Huang was supported by the NSF of China (Grant No. 12271457). The work of Shuyu Sun was supported by King Abdullah University of Science and Technology (KAUST) through the Grants BAS/1/1351-01 and URF/1/5028-01.

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian, 361005, China

    Huangxin Chen, Can Huang & Yahong Xiang

  2. Computational Transport Phenomena Laboratory, Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi Arabia

    Shuyu Sun

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  1. Huangxin Chen
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Correspondence to Shuyu Sun.

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Chen, H., Huang, C., Sun, S. et al. Stabilized Variational Formulations of Chorin-Type and Artificial Compressibility Methods for the Stochastic Stokes–Darcy Equations. J Sci Comput 101, 22 (2024). https://doi.org/10.1007/s10915-024-02663-6

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