FDTD: solving 1+1D delay PDE in parallel

TL;DR

This paper demonstrates a parallelizable finite-difference time-domain method for solving complex 1+1D delay PDEs in waveguide QED, enabling accurate multi-photon scattering simulations.

Contribution

It introduces a novel FDTD-based approach for delay PDEs with non-local terms, suitable for parallel computation and boundary condition implementation.

Findings

Successfully solves delay PDE with non-local terms

Enables parallel multi-threaded computation

Provides numerically exact multi-photon scattering solutions

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.