Spectral method for the unsteady incompressible Navier–Stokes equations in gauge formulation

A spectral method which uses a gauge method, as opposed to a projection method, to decouple the computation of velocity and pressure in the unsteady incompressible Navier–Stokes equations, is presented. Gauge methods decompose velocity into the sum of an auxilary field and the gradient of a gauge variable, which may, in principle, be assigned arbitrary boundary conditions, thus overcoming the issue of artificial pressure boundary conditions in projection methods. A lid-driven cavity flow is used as a test problem. A subtraction method is used to reduce the pollution effect of singularities at the top corners of the cavity. A Chebyshev spectral collocation method is used to discretize spatially. An exponential time differencing method is used to discretize temporally. Matrix diagonalization procedures are used to compute solutions directly and efficiently. Numerical results for the flow at Reynolds number Re = 1000 are presented, and compared to benchmark results. It is shown that the method, called the spectral gauge method, is straightforward to implement, and yields accurate solutions if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable.