  In superconducting circuit quantum information technologies, classical
microwave pulses are applied to control and measure the qubit states.
Currently, the design of these microwave pulses use simple theoretical or
numerical models that do not account for the self-consistent interactions of
how the qubit state modifies the applied microwave pulse. In this work, we
present the formulation and finite element time domain discretization of a
semiclassical Maxwell-Schr\"{o}dinger method for describing these
self-consistent dynamics for the case of a superconducting qubit capacitively
coupled to a general transmission line network. We validate the proposed method
by characterizing key effects related to common control and measurement
approaches for transmon and fluxonium qubits in systems that are amenable to
theoretical analysis. Our numerical results also highlight scenarios where
including the self-consistent interactions are essential. By treating the
microwaves classically, our method is substantially more efficient than
fully-quantum methods for the many situations where the quantum statistics of
the microwaves are not needed. Further, our approach does not require any
reformulations when the transmission line system is modified. In the future,
our method can be used to rapidly explore broader design spaces to search for
more effective control and measurement protocols for superconducting qubits.
