Convergence of gauge method for incompressible flow

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Abstract
A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a \mathbf {a} and a gauge variable ϕ \phi , u = a + ∇ ϕ \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 a \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 a \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 △ t / △ x ≤ C {\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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