Splitting and composition methods with embedded error estimators

Splitting and composition methods are by now standard numerical procedures to integrate differential equations in the realm of Geometric Numerical Integration, where preserving whatever invariants the systems has is of paramount importance. Nevertheless, even in problems where no qualitative properties have to be preserved and/or short time integrations are required, this class of methods have shown to be an excellent option when compared with other standard integrators.

It is in this setting where endowing splitting and composition methods with an efficient embedded error estimator could be most useful for step size control. The idea is to construct, in addition to the numerical solution, a second approximation from intermediate outputs, so that the difference is used as an estimator for the local error.

In this talk we show how these local order schemes can be obtained at each step as a linear combination of the intermediate stages of the integrator, so that the additional computational cost required for their evaluation is almost insignificant. The estimators thus constructed are subsequently used to adapt the step size along the integration.

Numerical examples show the efficiency of the procedure, in comparison with other well known embedded integration methods.