This paper deals with the application of the odd-even hopscotch (OEH) method for the numerical solution of the incompressible Navier-Stokes equations. The computation of the velocity and pressure is decoupled during the time integration in a predictor-corrector (PC) manner. This approach results in a Poisson equation for the pressure increment and necessitates the use of a fast Poisson solver for computational efficiency, since most of the CPU time is spent on the solution of this equation.
The author experiments with both an incomplete Choleski conjugate-gradient (ICCG) method and a multigrid (MG) method for solving the Poisson equation. The paper presents two numerical examples. The first, which has an exact solution, permits testing of the scheme’s accuracy, which is shown to be second order in space. The author demonstrates the superiority of the staggered grid over an ordinary grid for the velocity calculations. Viewed as an ordinary differential equation time-integration, the OEH-PC scheme is second order in time for velocity calculations but only first order for the computation of pressure.
In a second, more realistic example, the OEH-PC scheme is used to compute flow through a reservoir for various Reynolds numbers (100–800). The results are compared with an alternating-direction-implicit (ADI-PC) solution. As is well known, the ADI method is more stable than the OEH-PC scheme, which is only conditionally stable.
The paper is well presented and well written. The author does not, however, analyze the stability of the scheme in the presence of boundaries. In the comparison of fast Poisson solvers, the author considers only the ICCG and MG methods and finds the multigrid to be preferable to the ICCG for finer grids. Alternative adaptive and accelerated iterative schemes for the Poisson equation are not discussed. The OEH-PC is simpler to apply than the ADI-PC method, and its new application appears to be a valuable addition to the array of computational tools for solving the incompressible Navier-Stokes equations.
