On the implementation of the method of Magnus series for linear differential equations

The method of Magnus series has recently been analysed by Iserles and N phi rsett. It approximates the solution of linear differential equations y' = a(t)y in the form y(t) = e(sigma)(t)(yo), solving a nonlinear differential equation for a by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution. The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure. AMS subject classification;: 65L06, 34A50.