Robust Multigrid Algorithms for the Navier–Stokes Equations

Anisotropies occur naturally in computational fluid dynamics where the simulation of small-scale physical phenomena, such as boundary layers at high Reynolds numbers, causes the grid to be highly stretched, leading to a slowdown in convergence of multigrid methods. Several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation. However, they have rarely been applied to solving more complicated models, such as the incompressible Navier–Stokes equations. This paper contains the first published numerical results of the behavior of two popular robust multigrid approaches (alternating-plane smoothers combined with standard coarsening and plane-implicit smoothers combined with semi-coarsening) for solving the 3-D incompressible Navier–Stokes equations in the simulation of the driven-cavity and a boundary layer over a flat plate on a stretched grid. Grid size, grid stretching, and Reynolds number are the factors considered in evaluating the robustness of the multigrid methods. Both approaches yield large increases in convergence rates over cell-implicit smoothers on stretched grids. The combination of plane-implicit smoothers and semi-coarsening was found to be fully robust in the flat-plate simulation up to Reynolds numbers 106 and the best alternative in the driven-cavity simulation for Reynolds numbers above 103. The alternating-plane approach exhibits a better behavior for lower Reynolds numbers (below 103) in the driven-cavity simulation.