Analysis of Semi-Toeplitz Preconditioners for First-Order PDEs

Abstract

A semi-Toeplitz preconditioner for nonsymmetric, nondiagonally dominant systems of equations is studied. The preconditioner solve is based on a fast modified sine transform. As a model problem we study a system of equations arising from an implicit time discretization of a scalar hyperbolic partial differential equation (PDE). Analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system are derived. The convergence of a minimal residual iteration is shown to depend only on the spatial grid ratio and not on the number of unknowns.