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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
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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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: http://dx.doi.org/10.1090/S0025-5718-00-01248-5
PII: S 0025-5718(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