On Bi-grid Local Mode Analysis of Solution Techniques for 3-D Euler and Navier–Stokes Equations

Abstract

A procedure is presented for utilizing a bi-grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving Euler and Navier–Stokes equations. Model problems based on the convection equation, the diffusion equation, and Burger's equation are used to illustrate the superiority of the bi-grid analysis as a predictive tool for multigrid performance in comparison to the smoothing factor derived from conventional von Neumann analysis. For the Euler equations, bi-grid analysis is presented for three upwind difference based factorizations, namely spatial, eigenvalue, and combination splits, and two central difference based factorizations, namely LU and ADI methods. In the former, both the Steger–Warming and van Leer flux-vector splitting methods are considered. For the Navier–Stokes equations, only the Beam–Warming (ADI) central difference scheme is considered. In each case, estimates of multigrid convergence rates from the bi-grid analysis are compared to smoothing factors obtained from single-grid stability analysis. Effects of grid aspect ratio and flow skewness are examined. Both predictions are compared with practical multigrid convergence rates for 2-D Euler and Navier–Stokes solutions based on the Beam–Warming central difference scheme, and 3-D Euler solutions with various upwind difference schemes. It is demonstrated that bi-grid analysis can be used as a reliable tool for the prediction of practical multigrid performance.