Liang, Dong2015-08-282015-08-282014-04-072015-08-28http://hdl.handle.net/10315/29838Computation of Maxwell's equations has been playing an important role in many applications, such as the radio-frequencies, microwave antennas, aircraft radar, integrated optical circuits, wireless engineering and materials, etc. It is of particular importance to develop numerical methods to solve the equations effectively and accurately. During the propagation of electromagnetic waves in lossless media without sources, the energies keep constant for all time, which explains the physical feature of electromagnetic energy conservations in long term behaviors. Preserving the invariance of energies is an important issue for efficient numerical schemes for solving Maxwell's equations. In my thesis, we first develop and analyze the spatial fourth order energy-conserved splitting FDTD scheme for Maxwell's equations in two dimensions. For each time stage, while the spatial fourth-order difference operators are used to approximate the spatial derivatives on strict interior nodes, the important feature is that on the near boundary nodes, we propose a new type of fourth-order boundary difference operators to approximate the derivatives for ensuring energy conservative. The proposed EC-S-FDTD-(2,4) scheme is proved to be energy-conserved, unconditionally stable and of fourth order convergence in space. Secondly, we develop and analyze a new time fourth order EC-S-FDTD scheme. At each stage, we construct a time fourth-order scheme for each-stage splitting equations by converting the third-order correctional temporal derivatives to the spatial third-order differential terms approximated further by the three central difference combination operators. The developed EC-S-FDTD-(4,4) scheme preserves energies in the discrete form and in the discrete variation forms and has both time and spatial fourth-order convergence and super-convergence. Thirdly, for the three dimensional Maxwell's equations, we develop high order energy-conserved splitting FDTD scheme by combining the symplectic splitting with the spatial high order near boundary difference operators and interior difference operators. Theoretical analyses including energy conservations, unconditional stability, error estimates and super-convergence are established for the three dimensional problems. Finally, an efficient Euler-based S-FDTD scheme is developed and analyzed to solve a very important application of Maxwell's equations in Cole-Cole dispersive medium. Numerical experiments are presented in all four parts to confirm our theoretical results.enAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.MathematicsElectronic Thesis or Dissertation2015-08-28Energy-conservedS-FDTDSpatial fourth-orderMaxwell's equationsDispersive mediumCole–Cole model