Convergence of gauge method for incompressible flow

Wang, Cheng; Liu, Jian-Guo

doi:10.1090/S0025-5718-00-01248-5

Convergence of gauge method for incompressible flow
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by Cheng Wang and Jian-Guo Liu PDF

Math. Comp. 69 (2000), 1385-1407 Request permission

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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