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

Summary. 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.

[1]  R. Glowinski,et al.  Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem , 1977 .

[2]  Gilbert Strang,et al.  Accurate partial difference methods , 1964 .

[3]  E Weinan,et al.  Simple finite element method in vorticity formulation for incompressible flows , 2001, Math. Comput..

[4]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[5]  T. Taylor,et al.  Computational methods for fluid flow , 1982 .

[6]  Thomas Y. Hou,et al.  Convergence of a finite difference scheme for the Navier-Stokes equations using vorticity boundary conditions , 1992 .

[7]  Christopher R. Anderson,et al.  A High Order Explicit Method for the Computation of Flow About a Circular Cylinder , 1996 .

[8]  E Weinan,et al.  Essentially Compact Schemes for Unsteady Viscous Incompressible Flows , 1996 .

[9]  M Israeli,et al.  Numerical Simulation of Viscous Incompressible Flows , 1974 .

[10]  L. Quartapelle,et al.  Vorticity conditioning in the computation of two-dimensional viscous flows , 1981 .

[11]  Jian‐Guo Liu,et al.  Vorticity Boundary Condition and Related Issues for Finite Difference Schemes , 1996 .

[12]  S. Dennis,et al.  Compact h4 finite-difference approximations to operators of Navier-Stokes type , 1989 .

[13]  Chien Wang,et al.  Fourth order convergence of compact fi nite difference solver for 2-D incompressible fl ow , 2003 .

[14]  A. Thom,et al.  The flow past circular cylinders at low speeds , 1933 .

[15]  W. Roger Briley,et al.  A numerical study of laminar separation bubbles using the Navier-Stokes equations , 1971, Journal of Fluid Mechanics.

[16]  L. Quartapelle,et al.  Numerical solution of the incompressible Navier-Stokes equations , 1993, International series of numerical mathematics.

[17]  Heinz-Otto Kreiss,et al.  A fourth-order-accurate difference approximation for the incompressible Navier-Stokes equations☆ , 1994 .