A stable finite-difference time-domain scheme for local time-stepping on an adaptive mesh

Abstract

Physical effects driven by strong electromagnetic fields often occur in regions of highly localized fields on a scattering object. Unfortunately, the most common numerical technique for simulating time-domain electromagnetics, known as Finite-Difference Time-Domain (FDTD), is ill-equipped to handle such problems. A common solution to capture physics across many spatial scales is to use an adaptive mesh, which resolves temporal or spatial features exactly when and where they are needed, avoiding extra computation in space-time regions where it is unnecessary. We present a minimal modification to the FDTD algorithm that allows for a stable late-time solution to Maxwell's equations on an adaptive mesh with a Courant-Friedrichs-Levy limit of 5/6. An emphasis is placed on creating a simple, flexible, and easy to understand algorithm. The algorithm is implemented in 1D, 2D and 3D for geometries which are dynamic or possess large disparities in spatial or temporal scales. An example is presented which demonstrates the use of the algorithm in a resonant dielectric disc with a small slot.