Circulant Preconditioners for Solving Ordinary Differential Equations

In this paper, we consider the solution of ordinary diierential equations using boundary value methods. These methods require the solutions of one or more unsymmetric, large and sparse linear systems. Krylov subspace methods with the Strang block-circulant preconditioners are proposed for solving these linear systems. We prove that our preconditioners are invertible and all the eigenvalues of the preconditioned systems are clustered around 1. Therefore we expect fast convergence when Krylov subspace methods such as the GMRES method are applied to solving these preconditioned systems. Numerical results are reported to illustrate the eeectiveness of our methods.