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We have developed a new fully anisotropic 3D FDTD Maxwell solver forarbitrary electrically and magnetically anisotropic media for piecewiseconstant electric and magnetic materials that are co-located over the primarycomputational cells. Two numerical methods were developed that are callednon-averaged and averaged methods, respectively. The non-averaged method isfirst order accurate, while the averaged method is second order accurate forsmoothly-varying materials and reduces to first order for discontinuousmaterial distributions. For the standard FDTD field locations with theco-location of the electric and magnetic materials at the primary computationalcells, the averaged method require development of the different inversionalgorithms of the constitutive relations for the electric and magnetic fields.We provide a mathematically rigorous stability proof followed by extensivenumerical testing that includes long-time integration, eigenvalue analysis,tests with extreme, randomly placed material parameters, and various boundaryconditions. For accuracy evaluation we have constructed a test case with anexplicit analytic solution. Using transformation optics, we have constructedcomplex, spatially inhomogeneous geometrical object with fully anisotropicmaterials and a large dynamic range of $\underline{\epsilon}$ and$\underline{\mu}$, such that a plane wave incident on the object is perfectlyreconstructed downstream. In our implementation, the considerable increase inaccuracy of the averaged method only increases the computational run time by20%.

Stable 3D FDTD Method for arbitrary Fully Electric and Magnetic Anisotropic Maxwell Equations