  Josephson traveling-wave parametric amplifiers (JTWPAs) are wideband,
ultralow-noise amplifiers used to enable the readout of superconducting qubits.
While individual JTWPAs have achieved high performance, behavior between
devices is inconsistent due to wide manufacturing tolerances. Amplifier designs
could be modified to improve resilience towards variations in amplifier
components; however, existing device models often rely on analytical techniques
that typically fail to incorporate component variations. To begin addressing
this issue, a 1D numerical method for modeling JTWPAs is introduced in this
work. The method treats the Josephson junctions and transmission lines in an
amplifier as coupled subsystems and can easily incorporate arbitrary parameter
variations. We discretize the transmission line subsystem with a finite element
time domain method and the Josephson junction subsystem with a finite
difference method, with leapfrog time marching used to evolve the system in
time. We validate our method by comparing the computed gain to an analytical
model for a traditional JTWPA architecture and one with resonant phase
matching. We then use our method to demonstrate the impact of variations in
Josephson junctions and phase-matching resonators on amplification. In future
work, the method will be adjusted to incorporate additional amplifier
architectures and extended to a 3D full-wave approach.
