The Splitting High Order Finite-difference Time-domain Method For Maxwell's Equations In Two Dimensions

In this thesis, we consider high-order finite difference time domain methods (FDTD)of 2D Maxwell's equations by using operator-splitting and high-order (HO) finite difference.Two kinds of schemes, HO-S-FDTDI and HO-S-FDTDII are proposed and implemented.Theoretical analysis such as proof of stability and numerical dispersion analysisof the two schemes are given by Fourier methods . Numerical experiments are carriedout and confirm the theoretical analysis.The study is divided into five parts:Part I: An introduction to Maxwell's equations and related numerical methods aregiven.Part II:We consider high-order finite difference time domain methods of 2D Maxwell'sequations by using operator splitting and high-order finite difference methods. A newkind of splitting high-order finite difference time domain method on a staggered girdcalled HO-S- FDTDI and HO-S-FDTDII is firstly proposed, which consist of only twostages for each time step. By deriving their equivalent schemes we find that HO-S-FDTDIis of first order in time, fourth order in space ((1,4)- scheme), while HO-S-FDTDII issecond order in time and fourth order in space ((2,4)- scheme).Part III: By using the Fourier method, the two schemes are proved to be unconditionallystable and their numerical dispersion relations are derived. It is found thatHO-S-FDTDII has smaller dispersion error than HO-S-FDTDI, and that the former isnon-dissipative, the latter is dissipative. Numerical experiments are carried out and showHO-S-FDTDII is more accurate than HO-S-FDTDI.Part IV:We implement the boundary conditions of HO-S-FDTDI and HO-S-FDTDIIfor the two-dimensional Maxwell's equation in a rectangular domain, which is covered by perfectly electrical conductor. This part is complicated and needs hard work in processingthe discrete of the points near the boundary .Part V: We solve a rectangular wave guide problem by HO-S-FDTDI,HO-S- FDTDII andHO-ADI-FDTD. Numerical experiments show that the computational procedure of thetwo schemes is good stability and convergence , which verify the validity of the proposedschemes in simulating the waveguide problem.