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

Publication ,  Journal Article
Wang, C; Liu, JG
Published in: Numerische Mathematik
May 1, 2002

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. Thorn'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.

## Published In

Numerische Mathematik

0029-599X

May 1, 2002

91

3

## Start / End Page

543 / 576

• Numerical & Computational Mathematics
• 4904 Pure mathematics
• 4901 Applied mathematics
• 0103 Numerical and Computational Mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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Wang, C., & Liu, J. G. (2002). Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. Numerische Mathematik, 91(3), 543–576. https://doi.org/10.1007/s002110100311
Wang, C., and J. G. Liu. “Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation.” Numerische Mathematik 91, no. 3 (May 1, 2002): 543–76. https://doi.org/10.1007/s002110100311.
Wang C, Liu JG. Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. Numerische Mathematik. 2002 May 1;91(3):543–76.
Wang, C., and J. G. Liu. “Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation.” Numerische Mathematik, vol. 91, no. 3, May 2002, pp. 543–76. Scopus, doi:10.1007/s002110100311.
Wang C, Liu JG. Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. Numerische Mathematik. 2002 May 1;91(3):543–576.

## Published In

Numerische Mathematik

0029-599X

May 1, 2002

91

3

543 / 576