Title: FDTD: solving 1+1D delay PDE in parallel,
Abstract: We present a proof of concept for solving a 1+1D complex-valued, delay
partial differential equation (PDE) that emerges in the study of waveguide
quantum electrodynamics (QED) by adapting the finite-difference time-domain
(FDTD) method. The delay term is spatially non-local, rendering conventional
approaches such as the method of lines inapplicable. We show that by properly
designing the grid and by supplying the (partial) exact solution as the
boundary condition, the delay PDE can be numerically solved. In addition, we
demonstrate that while the delay imposes strong data dependency, multi-thread
parallelization can nevertheless be applied to such a problem. Our code
provides a numerically exact solution to the time-dependent multi-photon
scattering problem in waveguide QED.