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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field and a gauge variable , , 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 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 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 . 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:
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