Algebraic Smoothing Analysis of Multigrid Methods for the 2—D Compressible Navier-Stokes Equations

An algebraic smoothing analysis technique has been implemented to examine the performance of individual components in a multigrid method. The analysis leads to large-scale eigenvalue problems which are solved using Arnold!'s method. The analysis is applied to a semicoarsening multigrid method for solution of the 2-D compressible Navier-Stokes equations including a oneequation turbulence model. The governing equations are discretized using a second-order Roe flux-difference splitting method. On each grid level during the multigrid cycle, the solution is relaxed using a preconditioned multistage Runge-Kutta scheme. For the algebraic smoothing analysis, orthogonal projection operators of the multigrid restriction operator are constructed to decompose the solution error into fine- and coarsegrid modes. Using these projection operators, damping of the different error modes by preconditione d multistage relaxation on the finest grid is assessed and compared with design criteria. Scalar time-stepping, pointJacobi and implicit line-Jacobi preconditioners are examined. The performance of the entire multigrid cycle is also examined. The analysis is applied to two test cases: a inviscid, transonic channel flow and a high Reynolds number, turbulent, transonic airfoil. Results of the analyses for the preconditioned multistage relaxation schemes are consistent with Fourier analysis.

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