FDTD: Solving 1+1D delay PDE in parallel

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.本研究提出一种概念验证方案，通过适配时域有限差分法（FDTD），求解一类出现在波导量子电动力学（QED）研究中的1+1维复数值延迟偏微分方程（PDE）。该延迟项具有空间非局域性，使得线方法等传统数值方法无法直接适用。研究表明，通过合理设计计算网格，并将（部分）精确解作为边界条件施加，即可对该延迟偏微分方程进行数值求解。此外，本研究证实，尽管延迟项会带来极强的数据依赖性，但仍可针对该问题实现多线程并行计算。本研究配套的代码可对波导量子电动力学中的含时多光子散射问题给出数值精确解。