Implicit schemes for the integration of ODEs are popular when stability is more of concern than accuracy, for instance for the computation of a steady state solution. However, in particular for very large systems, the solution of the linear systems involved may be very expensive. When these systems are solved iteratively to a certain tolerance, it is often not known which tolerance has to be taken. We propose a different strategy, where the number of iterations is fixed, but the step size is controlled with respect to stability. Numerical tests show the effectiveness of this approach in comparison with an implicit scheme that iterates to a certain tolerance.
Bochev, M. A., Sleijpen, G. L. G., & van der Vorst, H. A. (1999). Stability control for approximate implicit time-stepping schemes with minimal residual iterations. Applied numerical mathematics, 31(3), 239-253. [10.1016/S0168-9274(98)00138-X]. https://doi.org/10.1016/S0168-9274(98)00138-X