Boundary and Interface Conditions for Electromagnetic Wave Propagation using FDTD

Abstract: Simulating electromagnetic waves is of increasing importance, for example, due to the rapidly growing demand of wireless communication in the fields of antenna design, photonics and electromagnetic compatibility (EMC). Many numerical and asymptotic techniques have been developed and one of the most common is the Finite-Difference Time-Domain (FDTD) method, also known as the Yee scheme. This centered difference scheme was introduced by Yee in 1966. The success of the Yee scheme is based on its relatively high accuracy, energy conservation and superior memory efficiency from the staggered form of defining unknowns. The scheme uses a structured Cartesian grid, which is excellent for implementations on modern computer architectures. However, the structured grid results in loss of accuracy due to general geometry of boundaries and material interfaces. A natural challenge is thus to keep the overall structure of Yee scheme while modifying the coefficients in the algorithm near boundaries and interfaces in order to improve the overall accuracy. Initial results in this direction have been presented by Engquist, Gustafsson, Tornberg and Wahlund in a series of papers. Our contributions are new formulations and extensions to higher dimensions. These new formulations give improved stability properties, suitable for longer simulation times. The development of the algorithmsis supported by rigorous stability analysis. We also tackle the problem of controlling the divergence free property of the solution—which is of extra importance in three dimensions—and present results of a number of numerical tests.