  Projection-based nonlinear model order reduction methods can be used to
reduce simulation times for the solution of many PDE-constrained problems. It
has been observed in literature that such nonlinear reduced-order models (ROMs)
based on Galerkin projection sometimes exhibit much larger stable time step
sizes than their unreduced counterparts. This work provides a detailed
theoretical analysis of this phenomenon for structural mechanics. We first show
that many desirable system matrix properties are preserved by the Galerkin
projection. Next, we prove that the eigenvalues of the linearized Galerkin
reduced-order system separate the eigenvalues of the linearized original
system. Assuming non-negative Rayleigh damping and a time integration using the
popular central difference method, we further prove that the theoretical linear
stability time step of the ROM is in fact always larger than or equal to the
critical time step of its corresponding full-order model. We also give
mathematical expressions for computing the stable time step size. Finally, we
show that under certain conditions this increase in the stability time step
even extends to some hyper-reduction methods. The findings can be used to
compute numerical stability time step sizes for the integration of nonlinear
ROMs in structural mechanics, and to speed up simulations by permitting the use
of larger time steps.
