The formulation of multiple-time-step integrators can provide substantial computational savings for mechanical systems with multiple time scales. However, the scope of these savings may be severely limited by the range of allowable time step choices. In this paper we have performed an exhaustive study of the linear stability of the fully asynchronous methods called AVI (asynchronous variational integrator), with two time steps, for essentially any combination of their values.

We have obtained approximate analytical expressions for the time step ratios that may render the scheme unstable in the case of linear equations, and verified them with extensive numerical computations. Synchronous multiple time stepping schemes such as r-RESPA show resonances when the outer step is a multiple of the effective half period of one of the fast oscillators. An elegant generalization is derived in the fully asynchronous case of AVI.