Adaptive time stepping for commutator free Lie group integrators

Abstract

Lie group integrators are steadily gaining popularity in many application areas. Until recently, there has been little work on Lie group integrators with adaptive time stepping. The most popular method for automatic step size selection seems in general to be that of embedded pairs of methods. For schemes that can be interpreted via local parameterisations, error estimation and step size selection can be determined in the parameter space. However, not all Lie group integrators fall into this class. Commutator-free Lie group integrators or Crouch-Grossman methods do not have such an interpretation. The problem can be resolved by using non-commutative order conditions. We develop new examples of embedded pairs of commutator-free Lie group integrators of orders 3(2) and 4(3), based on earlier work by Curry, C and Owren, B. (2019). Variable step size commutator-free Lie group integrators. Numer. Algorithms, 82(4), 1359-1376 where optimal pairs in terms of flow calculations and vector field evaluations per step were derived for orders 3(2) and 4(3). In particular we introduce a non-optimal embedded pair of orders 4(3) that actually behave better than the optimal ones for a popular test case, the heavy top.