Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Cheng Wang Indiana University Jian-Guo Liu University of Maryland

Numerical Analysis and Scientific Computing mathscidoc:1702.25055

Numerische Mathematik, 91, (3), 543–576, 2002.5
In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model.
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@inproceedings{cheng2002analysis,
  title={Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation},
  author={Cheng Wang, and Jian-Guo Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209112215150099385},
  booktitle={Numerische Mathematik},
  volume={91},
  number={3},
  pages={543–576},
  year={2002},
}
Cheng Wang, and Jian-Guo Liu. Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. 2002. Vol. 91. In Numerische Mathematik. pp.543–576. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209112215150099385.
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