A multigrid solver for the vorticity‐velocity Navier‐Stokes equations

This paper provides a multigrid incremental line-Gauss-Seidel method for solving the steady Navier-Stokes equations in two and three dimensions expressed in terms of the vorticity and velocity variables. The system of parabolic and Poisson equations governing the scalar components of the vector unknowns is solved using centred finite differences on a non-staggered grid. Numerical results for the two-dimensional driven cavity problem indicate that the spatial discretization of the equation defining the value of the vorticity on the boundary is extremely critical to obtaining accurate solutions. In fact, a standard one-sided three-point second-order-accurate approximation produces very inaccurate results for moderate-to-high values of the Reynolds number unless an exceedingly fine mesh is employed. On the other hand, a compact two-point second-order-accurate discretization is found to be always satisfactory and provides accurate solutions for Reynolds number up to 3200, a target impossible heretofore using this formulation and a non-staggered grid.