Math @ Duke

Publications [#246932] of JianGuo Liu
Papers Published
 Wang, C; Liu, JG, Convergence of gauge method for incompressible flow,
Mathematics of Computation, vol. 69 no. 232
(2000),
pp. 13851407
(last updated on 2018/03/24)
Abstract: A new formulation, a gauge formulation of the incompressible NavierStokes equations in terms of an auxiliary field a and a gauge variable φ, u = a + ∇φ, 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 CrankNicolson in time discretization. However, the boundary conditions for the auxiliary field 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 will be shown to be unconditionally stable for Stokes equations. For the full nonlinear NavierStokes equations the time stepping constraint is reduced to the standard CFL constraint Δt/Δx ≤ 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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