Volume 30, Issue 1
A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

Commun. Comput. Phys., 30 (2021), pp. 227-254.

Published online: 2021-04

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• Abstract

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.

65M06, 76D07

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@Article{CiCP-30-227, author = {Wang , Weiyi and Tan , Zhijun}, title = {A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {1}, pages = {227--254}, abstract = {

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0137}, url = {http://global-sci.org/intro/article_detail/cicp/18880.html} }
TY - JOUR T1 - A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids AU - Wang , Weiyi AU - Tan , Zhijun JO - Communications in Computational Physics VL - 1 SP - 227 EP - 254 PY - 2021 DA - 2021/04 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0137 UR - https://global-sci.org/intro/article_detail/cicp/18880.html KW - 3D immersed interface method, CG-Uzawa method, Stokes flow, level set method, staggered grids, singular forces. AB -

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.

Weiyi Wang & Zhijun Tan. (2021). A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids. Communications in Computational Physics. 30 (1). 227-254. doi:10.4208/cicp.OA-2020-0137
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