Reliable preconditioned iterative linear solvers for some numerical integrators

Implicit time-step numerical integrators for ordinary and evolutionary partial di erential equations need, at each step, the solution of linear algebraic equations that are unsymmetric and often large and sparse. Recently, a block preconditioner based on circulant approximations for the linear systems arising in the boundary value methods (BVMs) was introduced by the author. Here, some circulant approximations are compared and a further new type is considered. Numerical experiments are presented to check the e ectiveness of the various approximations that can be used in the underlying block preconditioner. Copyright ? 2001 John Wiley & Sons, Ltd.