Convergence of Gauge Methodfor Incompressible

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary eld a and a gauge variable , u = a + r, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and veriies 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 eld 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 eld a will be shown to be unconditionally stable for Stokes equations. For the full non-linear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint 4t=4x C. We also prove rst order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity eld is also obtained.