The Second-Order Accurate FDTD Equations In Inhomogeneous Media

The standard Finite Difference Time Domain (FDTD) method is second order accurate, but the accuracy will descend if the grids include the inhomogeneous media. Many efforts tried to choose proper effective permittivities to improve the accuracy of FDTD equations at dielectric interfaces. In two dimensional case, the second-order accurate FDTD equations on nonuniform lattices have been presented, but the accuracy of the equations were affected by the unsuitable mesh size in the regions in both sides of the interface.In this paper, an improvement of the placement of electric and magnetic field nodes is presented, and the rigorous second-order accurate FDTD equations of TE mode and TM mode at dielectric interface are given through the discretization of the integral forms of the Maxwell's curl equations and Taylor series extending of continuous field components. It is shown that the proper tangential and normal effective permittivities must be chosen on a non-uniform Yee's lattice available with an appropriate mesh size.The second order accurate FDTD equations at the dielectric interface in 3D case are analyzed. For the second order accuracy, the equations of distance parameter can be derived and the expression of effective permittivities can be obtained in the same manner. Then, a rectangular dielectric resonator is simulated.L₂ norm of the error is employed to calculate the error of the numerical solutions on nonuniform lattices. It is shown that the error based on the nonuniform lattices is less and linear convergent with the mesh size, and the resonator frequencies are computed in the method proposed in this paper with more accuracy than other methods.Finally, the second-order accurate FDTD equations in the grids filled with media of different permeabilities are considered. The proper tangential and normal effective permeabilities must be chosen on a non-uniform Yee's lattice available. The numerical results show that the validity and accuracy of the second-order accurate FDTD equations at magnetic media interface.