Time - symmetry and high - order Magnus methods

The subject of this paper is the investigation of the Magnus expansion of a solution of the linear diierential equation y 0 = a(t)y, y(0) 2 G, where G is a Lie group and a : R + ! g, g being the Lie algebra of G. We commence with a brief survey of recent work in this area. Next, building on earlier work of Iserles and NNrsett, we prove that an appropriate truncation of the expansion is time-symmetric. Moreover, we develop a 6th-order Lie-group solver based on the Magnus expansion which requires just three function evaluations and the calculation of seven commutators. The paper is accompanied by a number of numerical experiments.