An adaptive higher-order FDTD scheme for electromagnetic problems

Adaptive high-order FDTD schemes are developed to solve the Maxwell's equations with a bounded domain. Almost all derivatives in the Maxwell's equations are approximated by the higher order central-difference. Only second order approximation is implemented in the domain near the material discontinuities. Thus, on the domain away from the material boundary, the scheme is at least a fourth order in space and second order in time. This scheme uses the mesh stencil similar to the one used in the standard Yee cells and it is relatively easy to modify an existing code based on the Yee algorithm. Also, this scheme can be adapted for an unbounded space problem such as a scatter in an unbounded space. In this case, the Maxwell's equations are transformed to a set of auxiliary equations in a closed domain. A reflection-free amplitude-reduction scheme applied over the entire computational domain reduces the auxiliary field components outwardly and makes them equal to zero at the closed boundary. Since the relationship between the physical fields and their auxiliary counterparts is explicitly known and the former can be found from the latter with in the computational domain.