Locally-Refined and Semi-Conformal Mesh, Fully-Anisotropic Finite-Difference Time-Domain Solver

Abstract

We developed a new finite-difference time-domain (FDTD) Maxwell solver for simultaneously fully-anisotropic electric and magnetic media in three spatial dimensions. The algorithm's stability is proven mathematically and demonstrated numerically via eigenvalue analysis, showing that the neutral stability of the original Yee algorithm is preserved. We studied the algorithm's accuracy for fully-anisotropic dielectrics via a novel accuracy test utilizing metamaterial cloaks, a structure with known analytic solutions. We provide sufficient stability requirements for stable algorithms with diagonally-anisotropic and fully-anisotropic Drude models, developed as part of the transformation-based method—a method of incorporating non-uniform meshing into Maxwell solvers through the fully-anisotropic material parameters. For the transformation-based Maxwell FDTD solver, we have extended it to three spatial dimensions, outlined Gaussian local-mesh refinement, and introduced the semi-conformal mapping technique. Furthermore, we provide examples where each gridding technique exhibits an substantial increases in efficiency. The work lays the necessary foundation for the future of efficient and stable FDTD algorithms on structured meshes that, until now, have suffered from long-time instabilities.