# Finite difference vorticity methods

• Brian T. R. Wetton
Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1530)

## Abstract

Computations are presented for finite difference vorticity and primitive methods for the time dependent incompressible Navier Stokes equations in bounded domains. To directly compare solutions of the methods, modifications must be made to the primitive method and these are described. Both methods are based on semi-discrete methods that have been proven to be second order accurate. The computational results show second order uniform convergence in the velocities and vorticity using arbitrary (not compatible at time 0) initial data showing the validity of the time discrete methods for which only partial convergence results are known. An increase in the expected convergence order of the boundary vorticity using Thom's first order boundary condition is observed. Comparisons between the methods and between different boundary conditions in the vorticity method are given.

## Keywords

Discrete Fourier Transform Poiseuille Flow Order Convergence Periodic Channel Primitive Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
C.R. Anderson, “Derivation and Solution of the Discrete Pressure Equations for the Incompressible Navier-Stokes Equations,” preprint.Google Scholar
2. [2]
J. B. Bell, P. Colella and H. M. Glaz, “A Second-Order Projection Method for the Incompressible Navier-Stokes Equations,” J. Comput. Phys., 85, 257–283 (1989).
3. [3]
F.H. Harlow and J.E. Welsh, “Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surfaces,” Phys. Fluids, 8, 2181–2189 (1965).
4. [4]
J. G. Heywood and R. Rannacher, “Finite Element Approximation of the Nonstationary Navier-Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spacial Discretization,” SIAM J. Numer. Anal., 19, 275–311 (1982).
5. [5]
J. G. Heywood and R. Rannacher, “Finite Element Approximation of the Nonstationary Navier-Stokes Problem. II. Stability of Solutions and Error Estimates Uniform in Time.” SIAM J. Numer. Anal., 23, 750–777 (1986).
6. [6]
J. G. Heywood and R. Rannacher, “Finite Element Approximation of the Nonstationary Navier-Stokes Problem. IV. Error Analysis for Second Order Time Integration.” SIAM J. Numer. Anal., 27, 353–384 (1990).
7. [7]
T.Y. Hou and B.T.R. Wetton, “Convergence of a Finite Difference Scheme for the Navier Stokes Equations Using Vorticity Boundary Conditions,” to appear in the SIAM J. Numer. Anal., July, 1992.Google Scholar
8. [8]
T.Y. Hou and B.T.R. Wetton, “Second Order Convergence of a Projection Scheme for the Navier Stokes Equations with Boundaries,” submitted to the SIAM J. Number. Anal.Google Scholar
9. [9]
D. Michelson, “Convergence Theorems for Finite Difference Approximations for Hyperbolic Quasi-Linear Initial-Boundary Value Problems,” Math. Comp., 49, 445–459 (1987).
10. [10]
K. Meth, A Vortex and Finite Difference Hybrid Method to Compute the Flow of an Incompressible, Inviscid Fluid Past a Semi-infinite Plate, Thesis, New York University (1988).Google Scholar
11. [11]
M.J. Naughton, On Numerical Boundary Conditions for the Navier-Stokes Equations, Thesis, California Institute of Technology (1986).Google Scholar
12. [12]
S. A. Orszag and M. Israeli, “Numerical Simulation of Viscous Incompressible Flows,” Ann. Rev. of Fluid Mech., 6, 281–318 (1974).
13. [13]
R. Temam, “Behaviour at Time t=0 of the Solutions of Semi-linear Evolution Equations,” J. Diff. Eq., 43, 73–92 (1982).
14. [14]
F. V. Valz-Gris and L. Quartapelle, “Projection Conditions on the Vorticity in Viscous Incompressible Flows,” Int. J. Numer. Math. Fluids, 1, 453 (1981).
15. [15]
B.T.R. Wetton, Convergence of Numerical Approximations for the Navier-Stokes Equations with Boundaries: Vorticity and Primitive Formulations, Thesis, Courant Institute (1991).Google Scholar