  We present a well-posed constraint-preserving scheme for evolving first-order
metric perturbations on an arbitrary background with arbitrary source. We use
this scheme to evolve the leading-order metric perturbation in order-reduced
dynamical Chern-Simons gravity (dCS) on a Kerr background. In particular we
test the stability of stationary dCS data on a Kerr background with stationary
first-order dCS scalar field source. We find that the leading-order metric
perturbation numerically exhibits linear growth, but that the level of this
growth converges to zero with numerical resolution. This analysis shows that
spinning black holes in dCS gravity are numerically stable to leading-order
perturbations in the metric.
