Math @ Duke

Publications [#246937] of JianGuo Liu
Papers Published
 Wang, C; Liu, JG, Analysis of finite difference schemes for unsteady NavierStokes equations in vorticity formulation,
Numerische Mathematik, vol. 91 no. 3
(2002),
pp. 543576 [doi]
(last updated on 2017/12/13)
Abstract: In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible NavierStokes equations in vorticitystream function formulation. The noslip 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 longstencil 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 Strangtype 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 1D Stokes model.


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