  We have developed a new fully anisotropic 3D FDTD Maxwell solver for
arbitrary electrically and magnetically anisotropic media for piecewise
constant electric and magnetic materials that are co-located over the primary
computational cells. Two numerical methods were developed that are called
non-averaged and averaged methods, respectively. The non-averaged method is
first order accurate, while the averaged method is second order accurate for
smoothly-varying materials and reduces to first order for discontinuous
material distributions. For the standard FDTD field locations with the
co-location of the electric and magnetic materials at the primary computational
cells, the averaged method require development of the different inversion
algorithms of the constitutive relations for the electric and magnetic fields.
We provide a mathematically rigorous stability proof followed by extensive
numerical testing that includes long-time integration, eigenvalue analysis,
tests with extreme, randomly placed material parameters, and various boundary
conditions. For accuracy evaluation we have constructed a test case with an
explicit analytic solution. Using transformation optics, we have constructed
complex, spatially inhomogeneous geometrical object with fully anisotropic
materials and a large dynamic range of $\underline{\epsilon}$ and
$\underline{\mu}$, such that a plane wave incident on the object is perfectly
reconstructed downstream. In our implementation, the considerable increase in
accuracy of the averaged method only increases the computational run time by
20%.
