On the implementation of the method of Magnus series for linear differential equations (Q1293248)

This paper first reviews the solution of the matrix equation \(y'(H)= A(t)y(t)\) by the use of the Magnus expansion. Here the solution is written in the form \(y(t)= \exp(\Omega(H))y_0\), where \(\Omega\) is an infinite expansion of iterated commutators of the integrals of \(A(t)\), via a rooted tree theory approach. The authors then consider the error behaviour when this expansion is truncated and also give a two stage, fourth-order numerical method based on the evaluation of \(A\) at the Gaussian quadrature points. Some numerical results, but only for linear problems, are presented.