Volume 12, Issue 1
A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem

Adv. Appl. Math. Mech., 12 (2020), pp. 251-277.

Published online: 2019-12

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• Abstract

In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term $\theta\Delta \mathbf{u}$ in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter $\nu$, by choosing suitable parameter $\theta$. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition $\Delta\leq k_0$, $k_0>0$, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.

• Keywords

Navier-Stokes/Darcy equations, fractional step method, viscosity-splitting method, stability analysis, optimal error analysis.

76D05, 35Q30, 65M60, 65N30

wangyunxia@hpu.edu.cn (Yunxia Wang)

hanxuefeng@hpu.edu.cn (Xuefeng Han)

sizhiyong@hpu.edu.cn (Zhiyong Si)

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@Article{AAMM-12-251, author = {Wang , Yunxia and Han , Xuefeng and Si , Zhiyong }, title = {A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {251--277}, abstract = {

In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term $\theta\Delta \mathbf{u}$ in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter $\nu$, by choosing suitable parameter $\theta$. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition $\Delta\leq k_0$, $k_0>0$, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0084}, url = {http://global-sci.org/intro/article_detail/aamm/13426.html} }
TY - JOUR T1 - A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem AU - Wang , Yunxia AU - Han , Xuefeng AU - Si , Zhiyong JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 251 EP - 277 PY - 2019 DA - 2019/12 SN - 12 DO - http://dor.org/10.4208/aamm.OA-2019-0084 UR - https://global-sci.org/intro/article_detail/aamm/13426.html KW - Navier-Stokes/Darcy equations, fractional step method, viscosity-splitting method, stability analysis, optimal error analysis. AB -

In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term $\theta\Delta \mathbf{u}$ in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter $\nu$, by choosing suitable parameter $\theta$. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition $\Delta\leq k_0$, $k_0>0$, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.

Yunxia Wang, Xuefeng Han & Zhiyong Si. (2019). A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem. Advances in Applied Mathematics and Mechanics. 12 (1). 251-277. doi:10.4208/aamm.OA-2019-0084
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