A stable FDTD algorithm for non-diagonal, anisotropic dielectrics

A stable FDTD algorithm is developed for simulating Maxwell’s equations in anisotropic dielectric materials with principal axes not aligned with the grid. The algorithm is stable because the finite-difference operator that converts D to E is symmetric and positive semidefinite; for contrast, a previously developed asymmetric algorithm is shown to suffer from late-time instabilities. The presented algorithm has second-order error for continuous dielectric materials, and the error can be reduced to third-order by Richardson extrapolation. Applied to dielectrics with sharp interfaces, the algorithm has first-order error, even when averaging the dielectric within partially filled grid cells. However, averaging the dielectric permits Richardson extrapolation to obtain second-order error.