The behaviour of the local error in splitting methods applied to stiff problems

Splitting methods are frequently used in solving stiff differential equations, it is common to split the system of equations into a stiff and a nonstiff part. The classical theory for the local order of consistency is valid only for stepsizes which are smaller than what one would typically prefer to use in the integration. Error control and stepsize selection devices based on classical local order theory may lead to unstable error behaviour and inefficient stepsize sequences. Here, the behaviour of the local error in the Strang and Godunov splitting methods is explained by using two different tools, Lie series and singular perturbation theory. The two approaches provide and understanding of the phenomena from different points of view, but both are consistent with what is observed in numerical experiments.