STRANG-TYPE PRECONDITIONERS FOR SOLVING SYSTEM OF ODES BY BOUNDARY VALUE METHODS

In this paper, we survey some of the latest developments in using boundary value methods for solving systems of ordinary differential equations with initial values. These methods require the solutions of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type preconditioner is proposed for solving these linear systems. One of the main results is that if an A_{ν1, ν2}-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I + L where I is the identity matrix and the rank of L is at most 2m(ν₁+ν₂) .It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(ν₁+ν₂) + 1 iterations. Applications to differential algebraic equations and delay differential equations are also given.