On The Inhomogeneity Problems In The Finite-Difference Time-Domain Method

The finite-difference time-domain(FDTD) method is second-order accurate in homogeneous media and where the permittivity and permeability are smooth enough functions of coordinates.However,the across-interface difference equations are only first-order accurate in homogeneous-in-subregion media with a break at interface in permittivity or permeability,where the first inhomogeneity problem appears.The second inhomogeneity problem is the inhomogeneous or non-uniform grids in FDTD algorithm,where frner grids are used in sub-regions,in which electromagnetic waves vary fast with coordinates,in order to reduce the discrete errors.The third inhomogeneity problem arises in the perfectly matched layer(PML),a non-physical medium.Not only the physical medium but also the PML absorbing medium is objective medium when an open-region electromagnetic problem is simulated.The 1-,2-,and 3-dimensional homogeneous-in-sub-region media,where both the permittivity and permeability differ on different sides of the interface,are investigated respectively by using non-uniform grids,and the accuracy order of the across-interface difference equations is improved to be second-order through the rigorous mathematical deduction,based on which an idea of "electrical uniform" is abtained to be keeping the unitary-of-wavelength grid sizes uniform across the interface.As to problems where the traveling direction of waves could not be determined,it is demonstrated by numerical experiments that the errors in reflection coefficient using zero-angle-of-incidence non-uniform grids are almost identical to those using strict "electrical uniform" ones.Therefore,it is not necessary to be strictly "electrical uniform" for ordinary electromagnetic simulations.Instead,the grid sizes can be determined flexibly based on the "electrical uniform" method,which may be termed fuzzy "electrical uniform" method.Extending single direction across interface "electrical uniform" to betweendirection "electrical uniform",and then using the grids "electrical uniform" in both aspects,a precise finite-difference time-domain method can be obtained not only in homogeneous medium but also in homogeneous-in-subregion medium,with the time step size being maximum of the CFL stability criterion.This conclusion keeps correct for any of 1-dimensional,2-dimensional,and 3-dimensional cases.Whether the accuracy of the across-interface difference equations is first-order or second-order,the global accuracy keeps second-order.The accuracy order of reflection coefficient at the medium interface is different from that of the across-interface difference equations.For example,the reflection coefficient can be third-order accurate or even rigorously exact,but the accuracy order of the across-interface difference equations is preserved second-order.With the global accuracy being second-order always,the higher the accuracy at the medium interface,the smaller the global error.For the difficulty encountered in applying PML technique to an alternatingdirection implicit finite-difference time-domain(ADI-FDTD) method,two different ADI-FDTD formulas are introduced,both of which can decrease the reflection error by about 30dB.Because of their smaller reflection error,the new methods can be used for larger time steps(CFLN≤20)in the electromagnetic simulations of ADI-FDTD method.