# American Institute of Mathematical Sciences

• Previous Article
A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model
• ERA Home
• This Issue
• Next Article
The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions
September  2020, 28(3): 1191-1205. doi: 10.3934/era.2020065

## Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model

 Taiyuan University of Technology, Taiyuan 030024, China

* Corresponding author: jiahongen@aliyun.com

Received  March 2020 Revised  June 2020 Published  July 2020

Fund Project: The first author is supported by the Provincial Natural Science Foundation of Shanxi grant 201901D111123, Key Research and Development (R&D) Projects of Shanxi Province grant 201903D121038

In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

Citation: Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065
##### References:

show all references

##### References:
The global domain $\Omega$
Numerical results at time T = 1 for the standard scheme
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $|| e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.0158906 0.0352882 0.042823 $8$ 0.00847782 0.906408 0.0161872 1.12433 0.00978438 2.12983 $16$ 0.00436799 0.956724 0.00805077 1.00765 0.00230198 2.08761 $32$ 0.00221516 0.979559 0.00404369 0.993454 0.000609902 1.91623 $64$ 0.00111531 0.989966 0.00203112 0.993397 0.000131401 2.2146 $\frac{1}{h}$ $||e_\phi||_{L^2}$ $\phi_{L^2}rate$ $|| e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.0404764 0.0653031 0.500655 $8$ 0.0182477 1.14937 0.0182649 1.83808 0.25716 0.961151 $16$ 0.00927325 0.976568 0.00777467 1.23222 0.130569 0.977854 $32$ 0.00468612 0.984681 0.00370671 1.06864 0.0658343 0.987901 $64$ 0.00235584 0.992152 0.00183888 1.01131 0.0330473 0.994307
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $|| e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.0158906 0.0352882 0.042823 $8$ 0.00847782 0.906408 0.0161872 1.12433 0.00978438 2.12983 $16$ 0.00436799 0.956724 0.00805077 1.00765 0.00230198 2.08761 $32$ 0.00221516 0.979559 0.00404369 0.993454 0.000609902 1.91623 $64$ 0.00111531 0.989966 0.00203112 0.993397 0.000131401 2.2146 $\frac{1}{h}$ $||e_\phi||_{L^2}$ $\phi_{L^2}rate$ $|| e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.0404764 0.0653031 0.500655 $8$ 0.0182477 1.14937 0.0182649 1.83808 0.25716 0.961151 $16$ 0.00927325 0.976568 0.00777467 1.23222 0.130569 0.977854 $32$ 0.00468612 0.984681 0.00370671 1.06864 0.0658343 0.987901 $64$ 0.00235584 0.992152 0.00183888 1.01131 0.0330473 0.994307
Numerical results at time T = 1 for the standard grad-div scheme
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $||e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.015796 0.0349907 0.0332181 $8$ 0.00847477 0.898313 0.0161629 1.11429 0.00795638 2.06179 $16$ 0.00436788 0.956241 0.00804856 1.00588 0.00192546 2.04691 $32$ 0.00221515 0.979529 0.00404351 0.993123 0.000551883 1.80277 $64$ 0.00111531 0.98996 0.0020311 0.993347 0.000112974 2.28837 $\frac{1}{h}$ $|| e_\phi||_{L^2}$ $\phi_{L^2}rate$ $||e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.040389 0.0652755 0.512196 $8$ 0.018239 1.14694 0.0182613 1.83775 0.257443 0.992443 $16$ 0.00927265 0.975973 0.00777441 1.23198 0.130582 0.979297 $32$ 0.00468607 0.984603 0.00370669 1.0686 0.0658357 0.988014 $64$ 0.00235583 0.992143 0.00183888 1.0113 0.0330474 0.994333
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $||e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.015796 0.0349907 0.0332181 $8$ 0.00847477 0.898313 0.0161629 1.11429 0.00795638 2.06179 $16$ 0.00436788 0.956241 0.00804856 1.00588 0.00192546 2.04691 $32$ 0.00221515 0.979529 0.00404351 0.993123 0.000551883 1.80277 $64$ 0.00111531 0.98996 0.0020311 0.993347 0.000112974 2.28837 $\frac{1}{h}$ $|| e_\phi||_{L^2}$ $\phi_{L^2}rate$ $||e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.040389 0.0652755 0.512196 $8$ 0.018239 1.14694 0.0182613 1.83775 0.257443 0.992443 $16$ 0.00927265 0.975973 0.00777441 1.23198 0.130582 0.979297 $32$ 0.00468607 0.984603 0.00370669 1.0686 0.0658357 0.988014 $64$ 0.00235583 0.992143 0.00183888 1.0113 0.0330474 0.994333
Numerical results at time T = 1 for the modular grad-div scheme
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $|| e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.0161227 0.0422295 0.00546336 $8$ 0.00849478 0.924445 0.0186987 1.17531 0.00133666 2.03116 $16$ 0.00436888 0.959313 0.00849608 1.13807 0.000250008 2.41859 $32$ 0.00221526 0.979787 0.0042394 1.00294 7.36766e-005 1.7627 $64$ 0.00111531 0.990031 0.00204415 1.05236 6.93936e-006 3.40833 $\frac{1}{h}$ $|| e_\phi||_{L^2}$ $\phi_{L^2}rate$ $|| e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.0403803 0.0652523 0.500113 $8$ 0.0182335 1.14706 0.0182584 1.83747 0.257152 0.959633 $16$ 0.00927249 0.975563 0.00777436 1.23176 0.13057 0.977798 $32$ 0.00468608 0.984575 0.0037067 1.06859 0.0658345 0.987908 $64$ 0.00235583 0.992146 0.00183888 1.01131 0.0330473 0.994311
 $\frac{1}{h}$ $||e_{\bf u}||_{L^2}$ ${\bf u}_{L^2}rate$ $|| e_{\bf u}||_f$ ${\bf u}_{H_f}rate$ $||\nabla \cdot e_{\bf u}||_{L^2}$ $div{\bf u}_{L^2}rate$ $4$ 0.0161227 0.0422295 0.00546336 $8$ 0.00849478 0.924445 0.0186987 1.17531 0.00133666 2.03116 $16$ 0.00436888 0.959313 0.00849608 1.13807 0.000250008 2.41859 $32$ 0.00221526 0.979787 0.0042394 1.00294 7.36766e-005 1.7627 $64$ 0.00111531 0.990031 0.00204415 1.05236 6.93936e-006 3.40833 $\frac{1}{h}$ $|| e_\phi||_{L^2}$ $\phi_{L^2}rate$ $|| e_\phi||_p$ $\phi_{H_p}rate$ $|| e_p||_{L^2}$ $p_{L^2}rate$ $4$ 0.0403803 0.0652523 0.500113 $8$ 0.0182335 1.14706 0.0182584 1.83747 0.257152 0.959633 $16$ 0.00927249 0.975563 0.00777436 1.23176 0.13057 0.977798 $32$ 0.00468608 0.984575 0.0037067 1.06859 0.0658345 0.987908 $64$ 0.00235583 0.992146 0.00183888 1.01131 0.0330473 0.994311
The $||\nabla \cdot e_{\bf u}||_f$ for the standard without grad-div scheme, standard scheme and grad-div scheme with vaying hydraulic conductivity tensor $\mathbf K$
 ${\bf K}$ Non-stabilized Standard grad-div modular grad-div ${\bf I}$ 0.0169173 0.0108085 0.077557 $1e-1{\bf I}$ 0.0202974 0.0130437 0.0775562 $1e-2{\bf I}$ 0.0464625 0.0301879 0.077554 $1e-3{\bf I}$ 0.124238 0.0824988 0.0775521
 ${\bf K}$ Non-stabilized Standard grad-div modular grad-div ${\bf I}$ 0.0169173 0.0108085 0.077557 $1e-1{\bf I}$ 0.0202974 0.0130437 0.0775562 $1e-2{\bf I}$ 0.0464625 0.0301879 0.077554 $1e-3{\bf I}$ 0.124238 0.0824988 0.0775521
 [1] Tong Zhang, Jinyun Yuan. Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 849-865. doi: 10.3934/dcdsb.2014.19.849 [2] JaEun Ku. Maximum norm error estimates for Div least-squares method for Darcy flows. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1305-1318. doi: 10.3934/dcds.2010.26.1305 [3] Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109 [4] Zhipeng Yang, Xuejian Li, Xiaoming He, Ju Ming. A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021104 [5] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [6] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [7] Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 [8] Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497 [9] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [10] Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109 [11] Yann Brenier. Approximation of a simple Navier-Stokes model by monotonic rearrangement. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1285-1300. doi: 10.3934/dcds.2014.34.1285 [12] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [13] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [14] Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315 [15] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147 [16] Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41 [17] Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 [18] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [19] Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic & Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009 [20] Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115

2020 Impact Factor: 1.833