The Navier–Stokes-αβ equations as a platform for a spectral multigrid method to solve the Navier–Stokes equations

This paper describes a spectral multigrid method for spatially periodic homogeneous and isotropic turbulent flows. The method uses the Navier–Stokes-αβ equations to accelerate convergence toward solutions of the Navier–Stokes equations. The Navier–Stokes-αβ equations are solved on coarse grids at various levels and the Navier–Stokes equations are solved on the “nest grid”. The method uses Crank–Nicolson time-stepping for the viscous terms, explicit time-stepping for the remaining terms, and Richardson iteration to solve linear systems encountered at each time step and on each grid level. To explore the computational efficiency of the method, comparisons are made with results obtained from an analogous spectral multigrid method for the Navier–Stokes equations. These comparisons are based on computing work units and residuals for multigrid cycles. Most importantly, we examine how choosing different values of the length scales α and β entering the Navier–Stokes-αβ equations influence the efficiency and accuracy of these multigrid schemes.