The Importance of Eigenvectors for Local Preconditioners of the Euler Equations

Most previous preconditioning efforts have focused on manipulating the eigenvalues of the spatial operator. For The design of local preconditioners to accelerate the convergence to a steady state for the compressible Euler equations has so far example, Turkel [2] derives a family of preconditioners been solely based on eigenvalue analysis. However, numerical eviwhich reduces the spread of the wave speeds for pseudodence exists that the eigenvector structure also has an influence compressible and low Mach number compressible flows. on the performance of preconditioners and should therefore be In [3], van Leer et al. derive a symmetric preconditioner included in the design process. In this paper, we present the mathefor the two-dimensional Euler equations which reduces matical framework for the eigenvector analysis of local preconditioners for the multi-dimensional Euler equations. The non-normalthe spread of the characteristic speeds across from (M 1 ity of the preconditioned system is crucial in determining the 1)/min(M, uM 2 1u) to 1/Ï1 2 min(M2, M22), where M is potential for transient amplification of perturbations. Several exthe Mach number. Lee [9] shows that this is the lowest isting local preconditioners are shown to possess a highly nonratio of characteristic speeds attainable using a symmetric normal structure for low Mach numbers. This non-normality leads preconditioner. For grid-aligned upwind schemes, Allto significant robustness problems at stagnation points. A modification to these preconditioners which eliminates the non-normality is maras [6] finds that a block Jacobi preconditioner clusters suggested, and numerical results are presented showing the marked the high frequency eigenmodes of the two-dimensional improvement in robustness. Q 1996 Academic Press, Inc. Euler and Navier–Stokes discretized operator allowing the formulation of an effective smoother for multigrid algorithms. Although the block Jacobi preconditioner does

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