FDTD: solving 1+1D delay PDE

We present a proof of concept for adapting the finite-difference time-domain method (FDTD) for solving a 1+1D complex-valued, delay partial differential equation (PDE) that emerges in the study of waveguide quantum electrodynamics (QED). The delay term exists in both spatial and temporal directions, rendering the conventional approaches such as the method of lines inapplicable. We show that by properly designing the grid and by using the exact (partial) solution as the boundary condition, the delay PDE can be numerically solved. Our code provides a numerically exact solution to the time-dependent multi-photon scattering problem in waveguide QED. The program is written in C and open-sourced on GitHub.