The Fer expansion and time symmetry: a Strang-type approach

We introduce a symmetric version of the Fer expansion for the solution of the ODE y 0 = (t; y)y. We show that the scheme are time symmetric for linear problems, and, because of the symmetry of the scheme, one gets order 2p + 2 whereby the classical Fer expansion would display order 2p + 1 only. This reduces signiicantly the computational cost of the classical Fer expansion. We prove the convergence of the proposed symmetric schemes and obtain an estimate of the radius of convergence. We present some numerical methods of this type and test them on a linear and a nonlinear problem. The numerical experiment connrm that the proposed methods are time-symmetric to machine accuracy for linear problem, while, for nonlinear problem, they are not time symmetric but obey time-symmetry to a higher order of accuracy. 1 The classical Fer expansion We consider the diierential equation y 0 = (t; y)y; y(0) = y 0 ; (1.1) where y 2 R nn and (t; y) is a n n matrix function of y and, possibly, of the time t. Of particular relevance is the case when y evolves on a matrix Lie group G, in which case 2 g, the Lie algebra of G (see 14, 13, 10, 19] and references therein), in which case it is useful that also (numerical) approximations of the exact solution y(t) remain on the same Lie group G. Such Lie-group approximations can be obtained by means of the Fer expansion 19, 20], the Magnus expansion and many more exponential expansions. In particular, the Fer expansion consists of writing the solution y(t) of (1:1) as the product of innnite exponentials, i.e. a procedure that is equivalent to the following algorithm: Write y(t) = e 0 r 1 (t)y 0 , where 0 is the integral of a known function, that we choose to be 0 (t) = Z t 0 (s; y) ds (1.3) (note the dependence of the exact solution y of (1:1)). We derive the diierential equation obeyed by the rst residual r 1 : from