Block {ω}-circulant preconditioners for the systems of differential equations

The numerical solution of large and sparse nonsymmetric linear systems of algebraic equations is usually the most time consuming part of time-step integrators for differential equations based on implicit formulas. Preconditioned Krylov subspace methods using Strang block circulant preconditioners have been employed to solve such linear systems. However, it has been observed that these block circulant preconditioners can be very ill-conditioned or singular even when the underlying nonpreconditioned matrix is well-conditioned. In this paper we propose the more general class of the block { ω }-circulant preconditioners. For the underlying problems, ω can be chosen so that the condition number of these preconditioners is much smaller than that of the Strang block circulant preconditioner (which belongs to the same class with ω =1) and the related iterations can converge very quickly.