Analysis of Robust Multigrid Methods for Steady Viscous Low Mach Number Flows

A theoretical and numerical analysis of different implicit methods in multigrid form is carried out in order to study their behaviour at various flow conditions. The Fourier analysis for scalar convection?diffusion equations is extended to the coupled set of laminar Navier?Stokes equations. For first- and second-order discretized Navier?Stokes equations, the analysis of point, line, and block methods is in good agreement with the numerical results. The influence on the convergence rate of different second-order implementations in the multigrid method is also considered. The combination of multigrid, line-methods, and mixed discretization enables to tackle stiff problems in a cheap and robust way. The comparison is further extended to Reynolds averaged Navier?Stokes equations with the low-Reynoldsk-? equations describing the turbulence. When line and block methods are used, it is advantageous not to incorporate the turbulence equations into the multigrid cycle.