  We present a numerical method to approximate the long-time asymptotic
solution $\rho_\infty(t)$ to the Lindblad master equation for an open quantum
system under the influence of an external drive. The proposed scheme uses
perturbation theory to rank individual drive terms according to their dynamical
relevance, and adaptively determines an effective Hamiltonian. In the
constructed rotating frame, $\rho_\infty$ is approximated by a
time-independent, nonequilibrium steady-state. This steady-state can be
computed with much better numerical efficiency than asymptotic long-time
evolution of the system in the lab frame. We illustrate the use of this method
by simulating recent transmission measurements of the heavy-fluxonium device,
for which ordinary time-dependent simulations are severely challenging due to
the presence of metastable states with lifetimes of the order of milliseconds.
