Numerische Mathematik

, Volume 91, Issue 3, pp 543–576 | Cite as

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

  • Cheng Wang
  • Jian-Guo Liu
Original article


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.

Mathematics Subject Classification (1991): 65M06, 76M20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Cheng Wang
    • 1
  • Jian-Guo Liu
    • 2
  1. 1.Institute for Scientific Computing and Applied Mathematics and Department of Mathematics, Indiana University, Bloomington, IN 47405, USA; e-mail: US
  2. 2.Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, MD 20742, USA; e-mail: US

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