Geometric integrators for multiple time-scale simulation

In this paper, we review and extend recent research on averaging integrators for multiple time-scale simulation such as are needed for physical N-body problems including molecular dynamics, materials modelling and celestial mechanics. A number of methods have been proposed for direct numerical integration of multiscale problems with special structure, such as the mollified impulse method (Garcia-Archilla, Sanz-Serna and Skeel 1999 SIAM J. Sci. Comput. 20 930-63) and the reversible averaging method (Leimkuhler and Reich 2001 J. Comput. Phys. 17195-114). Features of problems of interest, such as thermostatted coarse-grained molecular dynamics, require extension of the standard framework. At the same time, in some applications the computation of averages plays a crucial role, but the available methods have deficiencies in this regard. We demonstrate that a new approach based on the introduction of shadow variables, which mirror physical variables, has promised for broadening the usefulness of multiscale methods and enhancing accuracy of or simplifying computation of averages. The shadow variables must be computed from an auxiliary equation. While a geometric integrator in the extended space is possible, in practice we observe enhanced long-term energy behaviour only through use of a variant of the method which controls drift of the shadow variables using dissipation and sacrifices the formal geometric properties such as time-reversibility and volume preservation in the enlarged phase space, stabilizing the corresponding properties in the physical variables. The method is applied to a gravitational three-body problem as well as a partially thermostatted model problem for a dilute gas of diatomic molecules.