The importance of eigenvectors for local preconditioners of the euler equations

The design of local preconditioners to accelerate the convergence to a steady state for the compressible Euler equations has so far been solely based on eigenvalue analysis. However, numerical evidence exists that the eigenvector structure also has an influence on the performance of preconditioners, and should therefore be included in the design process. In this paper, we present the mathematical framework for the eigenvector analysis of local preconditioners for the multi-dimensional Euler equations. The non-orthogonality of the preconditioned system is crucial in determining the potential for transient amplification of perturbations. Several existing local preconditioners are shown to possess a highly non-normal structure for low Mach numbers. This nonnormality leads to significant robustness problems at stagnation points. A modification to these preconditioners which eliminates the non-normality is suggested, and numerical results are presented showing the marked improvement in robustness.