Analysis of Preconditioners for Hyperbolic Partial Differential Equations

Abstract

Preconditioners for conjugate gradient (CG)-like iterative methods are analyzed. The systems of equations arise from discretizations of time-dependent hyperbolic partial differential equations (PDEs) in two space dimensions. Incomplete LU (ILU) and block ILU preconditioners are considered for a model problem subject to periodic boundary conditions. The spectra of the preconditioned coefficient matrices are determined by Fourier analysis. It is found that the condition number is not improved by the ILU preconditioners. Furthermore, the number of distinct eigenvalues is not decreased. A semicirculant preconditioner applied to a problem, subject to Dirichlet boundary conditions at the inflow boundaries, is also examined. Analytical formulas for the eigenvalues and the eigenvectors are derived. For this purpose a special analysis, based on spectral decomposition and residue theory, is developed. When the grid ratio in space is less than one, the spectrum asymptotically becomes two finite curve segments, which are independent of the number of gridpoints. For the restarted generalized minimal residual (GMRES) iteration, a slight reduction of the grid ratio from one substantially improves the convergence rate. This is also predicted by an asymptotic analysis.