In this talk, the Strang block-circulant preconditioner is proposed to solve the linear systems arising from the numerical method of ordinary differential equations (ODEs). The linear multistep methods are taken into account here. These implicit numerical methods for solving ODEs require the solution of one or more unsymmetric, large and sparse linear systems at each integration step. Therefore, iterative methods such as the Krylov subspace methods, can be employed to solving these linear systems. We will prove that if the A_k1,k2-stable boundary value method is used to discretize the m-by-m system of ODEs, then our proposed block-circulant preconditioners are invertible and all the eigenvalues of the preconditioned system are 1 except for at most 2m(k1+k2) outliers that are independent of the integration step size. It follows that when the Krylov subspace method (for instance, the GMRES method) is applied to solving these preconditioned systems, we can expect a fast convergence rate. Numerical results are reported to illustrate the effectiveness of our methods.

MINISIMPOSIUM SESSION: 9. Solving linear and nonlinear systems in differential equations