Improved high order integrators based on the Magnus expansion (Q1587307)

This paper presents high-order efficient numerical integration methods for solving the initial value problem defined by the linear differential equation \[ \dot X= A(t)X,\qquad X(t_0)=\dot I, \] based on the Magnus expansion. These methods preserve intrinsic geometric properties of the exact solution. New approximation schemes of order four, six and eight to the Magnus expansion involving exclusively single analytical integrals with just one, four and ten commutators, respectively, are obtained. Sixth- and eight-order numerical algorithms with automatic step-size control are constructed explicitly. The theory of free Lie algebra is used.