Stability control for approximate implicit time­stepping schemes with minimal residual iterations

Implicit schemes for the integration of ODE's are popular when stabil­ ity is more of concern than accuracy, for instance for the computation of a steady state solution. However, in particular for very large sys­ tems the solution of the involved linear systems may be very expensive. In this paper we study the solution of these linear systems by a mod­ erate number of iterations of the minimum residual iterative method GMRES. Of course, this puts limits to the step size since these ap­ proximate schemes may be viewed as explicit schemes and these are never unconditionally stable. It turns out that even a modest degree of approximation allows rather large time steps and we propose a simple mechanism for the control of the step size with respect to stability