The Splitting High Order Finite-Difference Time-Domain Methods For Maxwell's Equations In Two Dimensions

In this thesis, we study ?nite di?erence methods (FD) of Maxwell's equations byusing operator-splitting and their application in computation. Firstly, we propose high-order symmetrical ?nite di?erence time domain methods (FDTD) for the 2D Maxwell'sequations by using operator-splitting and high-order (HO) ?nite di?erence methods.Then, theoretical analysis of HO-SS-FDTD on stability and numerical dispersion er-ror is given by Fourier methods. Numerical experiments are carried out and con?rm thetheoretical analysis. Thirdly, we consider further analysis of the symmetrical splitting?nite di?erence time domain methods (SS-FDTD) of 3D Maxwell's equations. We drivenew energy conserved identities of this scheme and give stability analysis. Numericalexperiments are presented and computational results show that the SS-FDTD scheme isenergy conserved in the discrete H1 norm.The study is divided into three parts:Part I: We introduce the background and importance of selected research topic, in-cluding the model equations of the research problem and some usual numerical methodsof these equations.Part II: We consider high-order ?nite di?erence time domain methods of 2D Maxwell'sequations by using operator splitting and high-order ?nite di?erence methods. A newkind of symmetric splitting high-order ?nite di?erence time domain method for the 2DMaxwell's equations, called HO-SS-FDTD, is ?rstly proposed. This method using theYee's staggered grids consists of four stages of equations, and can be solved e?ciently. By deriving its equivalent scheme we ?nd that HO-SS-FDTD is of second order in time,fourth order in space (denoted by (2,4)- scheme). By using the Fourier method, thismethods is proved to be unconditionally stable and has reasonable numerical dispersionerror. Numerical dispersion relation of HO-SS-FDTD is derived and compared with theCrank-Nicolson scheme, CN-FDTD. It is found that HO-SS-FDTD is non-dissipative.Numerical experiments are carried out and show that the variation of the module of thegrowth factor. Numerical dispersion error is illustrated in the di?erent cases. Finally, Weimplement the boundary conditions of HO-SS-FDTD for the two-dimensional Maxwell'sequation in a rectangular domain with the perfectly electrical conducting boundary condi-tions. This part is useful in processing the discretization of the points near the boundaryand in programming.Part III: We study the conservation of energy of the symmetric splitting ?nite-di?erence time-domain(SS-FDTD) methods for the 3D Maxwell s equations with zeroconductivity(σ= 0). By using new energy methods and the operated SS-FDTD schemesby the di?erence operatorsδx,δy,δz, we derive the energy conserved identities SS-FTDTin the discrete H1-norm, which show that SS-FDTD is conserved in the discrete H1 norm.Numerical experiments are done and verify the energy conservation of the solution .