Adjoint and selfadjoint Lie-group methods

In the past few years, a number of Lie-group methods based on Runge-Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge-Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie-group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness. In this paper we discuss Lie-group methods and their dependence on centering coordinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge-Kutta schemes and symmetric coordinates, based on a geodesic or on a flow midpoint. As a result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.