Performance of geometric multigrid method for coupled two-dimensional systems in CFD

The performance of a geometric multigrid method is analyzed for two-dimensional Laplace, Navier, Burgers and two formulations of Navier–Stokes (streamfunction–vorticity and streamfunction–velocity) equations. These equations are discretized with the Finite Difference Method on uniform grids with numerical approximations of first- and second-orders of accuracy. The systems of equations are solved with a Modified Strongly Implicit (MSI) and a Successive Over Relaxation (SOR) solver associated with the multigrid method with a V-cycle and a Correction Scheme (CS) and a Full Approximation Scheme (FAS). The effect of the number of inner iterations of the solver, the number of grid levels problems with grid sizes of 1025 × 1025 points, the influence of differential equations numbers and Reynolds number up to 1000 on Central Processing Unit (CPU) time are investigated. The results show that (1) a solution of two coupled equations (Navier or Burgers) is obtained with the same efficiency multigrid textbook that occurs in the solution of only one equation (Laplace), (2) the efficiency of the multigrid method in the solution of two coupled equations (Navier–Stokes streamfunction–vorticity formulation) or only one equation (the Navier–Stokes streamfunction–velocity formulation) decreases with increasing Reynolds numbers, and (3) the poor performance of the multigrid method for solving the Navier–Stokes seems to be related to the physics of the problem and not to the type of formulation or coupling between the equations.