Strang-type preconditioners for systems of LMF-based ODE codes

We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if an A(k1,k2)-stable boundary value method is used for an m-by-m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k(1) + k(2)) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k(1) + k(2)) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.