Analogous to RKF variable step size methods. Is there a way to develop a variable step size scheme for an implicit integrator.
Thanks a lot for your answer.
I think you are talking about the error in evaluating the state using different step size schemes. This does not relate to the error estimate as far as I know as unlike explicit RK method the step size is not related to the true solution of the diff eq.
In other words, in RK methods when the step size is reduced the solution accuracy is improved according to a certain order (error estimate). My question is concerned with implicit methods and if such a relation exists or not.
Of course-the fact that the method is implicit is irrelevant. Whether explicit or implicit, what such methods provide is the one-step transformation that acts on the state variable(s) at step t and provides the value of the state variable(s) at time t+Δt. This transformation is a function of the step size.
By monitoring the value of certain quantities an estimate of the error associated with the step size choice can be found and the step size changed accordingly. It doesn't matter whether it's implicit or explicit.
Incidentally, it's not true that accuracy is, necessarily, improved, as one decreases step size. The reason is that, as the step size decreases, the number of steps necessary to reach a given physical time, increases-and individual errors can accumulate.
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