A Nonlinear Multigrid Method for the Three-Dimensional Incompressible Navier–Stokes Equations

Abstract

A nonlinear multigrid method is developed for solving the three-dimensional Navier–Stokes equations in conjunction with the artificial compressibility formulation. The method is based on the full multigrid (FMG)—full approximation storage (FAS)—algorithm and is realized via an “unsteady-type” procedure, according to which the equations are not solved exactly on the coarsest grid, but some pseudo-time iterations are performed on the finer grids and some on the coarsest grid. The multigrid method is implemented in conjunction with a third-order upwind characteristics-based scheme for the discretization of the convection terms, and the fourth-order Runge–Kutta scheme for time integration. The performance of the method is investigated for three-dimensional flows in straight and curved channels as well as flow in a cubic cavity. The multigrid acceleration is assessed in contrast to the single-grid and mesh-sequencing algorithms. The effects of various multigrid components on the convergence acceleration, such as prolongation operators, as well as pre- and postrelaxation iterations, are also investigated.