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Convergence of gauge method for incompressible flow


Authors: Cheng Wang and Jian-Guo Liu
Journal: Math. Comp. 69 (2000), 1385-1407
MSC (1991): Primary 65M12, 76M20
DOI: https://doi.org/10.1090/S0025-5718-00-01248-5
Published electronically: March 24, 2000
MathSciNet review: 1710695
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Abstract:

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field $\mathbf{a}$ and a gauge variable $\phi$, $\mathbf{u} =\mathbf{a}+\nabla\phi$, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field $\mathbf{a}$ are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field $\mathbf{a}$ will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint ${\triangle t} / {\triangle x} \le C$. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.


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  • 1. A.S. Almgren, J.B. Bell, and W.G. Szymczak, A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput. 17 (1996), 358-369. MR 96j:76104
  • 2. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745-762. MR 39:3723
  • 3. J.B. Bell, P. Colella, and H.M. Glaz, A second order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys. 85 (1989), 257-283. MR 90i:76002
  • 4. T. Buttke, Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow, Vortex Flows and Related Numerical Methods, (Grenoble, 1992; J. T. Beale et al., eds.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 395, Kluwer, Dordrecht, 1993, pp. 39-57. MR 94m:760093
  • 5. T. Buttke and A. Chorin, Turbulence calculation using magnetization variables, Appl. Numer. Math. 12 (1993), 47-54. MR 94e:76016
  • 6. Weinan E and J.-G. Liu, Gauge method for viscous incompressible flows, submitted to J. Comp. Phys. (1996)
  • 7. Weinan E and J.-G. Liu, Gauge finite element method for incompressible flows, to appear in Int. J. Num. Meth. Fluids (2000).
  • 8. Weinan E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), 1017-1057. Projection method III: Spatial discretizations on the staggered grid, to appear Math. Comp. MR 96e:65061
  • 9. Weinan E and J.-G. Liu, Finite difference schemes for incompressible flows in the impulse-velocity formulation, J. Comput. Phys. 130 (1997), 67-76. MR 97j:76036
  • 10. P.M. Gresho and R.L. Sani, On pressure boundary conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 7 (1987), 1111-1145.
  • 11. G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, New York (1999)
  • 12. J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comp. Phys. 59 (1985), 308-323. MR 87a:76046
  • 13. J.H. Maddocks and R.L. Pego, An unconstrained Hamiltonian formulation for incompressible fluid, Comm. Math. Phys. 170 (1995), 207-217. MR 96a:76085
  • 14. S.A. Orszag, M. Israeli, and M.O. Deville, Boundary conditions for incompressible flows, J. Scientific Computing 1 (1986), 75-111.
  • 15. V.I. Oseledets, On a new way of writing the Navier-Stokes equation. The Hamiltonian formulation, J. Russ. Math. Surveys 44 (1989), 210-211. MR 91e:58173
  • 16. G. Russo and P. Smereka, Impulse formulation of the Euler equations: general properties and numerical methods, J. Fluid Mech. 391 (1999), 189-209. CMP 99:17
  • 17. J. Shen, On error estimates of projection methods for Navier-Stokes equation: first order schemes, SIAM J. Numer. Anal. 29 (1992), 57-77. MR 92m:35213
  • 18. J. Shen, On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), 49-73. MR 93a:35122
  • 19. G. Strang, Accurate partial differential methods II. Non-linear problems, Numer. Math. 6 (1964), 37-46. MR 29:4215
  • 20. R. Temam, Sur l'approximation de la solution des equation de Navier-Stokes par la méthode des fractionnaires II, Arch. Rational Mech. Anal. 33 (1969), 377-385. MR 39:5968
  • 21. J. van Kan, A second order accurate pressure correction scheme for viscous incompressible flow, SIAM J. Sci. Comput. 7 (1986), 870-891. MR 87h:76008
  • 22. B. R. Wetton, Error analysis for Chorin's original fully discrete projection method and regularizations in space and time, SIAM J. Numer. Anal. 34 (1997), 1683-1697. MR 95c:65161

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Additional Information

Cheng Wang
Affiliation: Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: cwang@math.umd.edu

Jian-Guo Liu
Affiliation: Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: jliu@math.umd.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01248-5
Keywords: Viscous incompressible flows, gauge method, convergence, explicit boundary condition
Received by editor(s): November 10, 1997
Received by editor(s) in revised form: December 7, 1998
Published electronically: March 24, 2000
Additional Notes: The research was supported by NSF grant DMS-9805621 and Navy ONR grant N00014-96-1013
Article copyright: © Copyright 2000 American Mathematical Society

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