Stability control for approximate implicit time-stepping schemes with minimal residual iterations

Abstract

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 maybevery 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 approximationallows rather large time steps and we propose a simple mechanism for the control of the step size with respect to stability.