Finite-Difference Time-Domain Method For Maxwell's Equations In Complex Dispersive Medium:Analysis And Applications

In electromagnetics,if a medium's physical parameters(permittivity or per-meability)depends on the wave frequency,then this medium is called dispersive medium.Electromagnetic waves with different frequencies in dispersive medium travel at different velocities,we call this phenomenon dispersion.In fact,most substances belong to dispersive medium,such as soil,water,plasma,biological tissues,and so on.Due to the complex electromagnetic behavior of wave in dis-persive medium,it is becoming more and more important to study the numerical methods of Maxwell's equations in complex dispersive medium.This thesis is mainly focused on the analysis and application of the Finite-Difference Time-Domain method(FDTD)of Maxwell's equations in complex dispersive medium.In this thesis,we fist,study some now results of FDTD method for solving the metamaterial Maxwell's equations on non-uniform rectangular meshes.After-wards,we mainly focus on the FDTD method for solving the Maxwell's equations in a Cole-Cole dispersive medium,and propose three numerical method,then pro-vided rigorous theoretical analysis of stability and convergence.Meanwhile,we construct two efficient FDTD algorithm,and give the detailed derivation process and implementation details.Finally,the numerical results verified the accuracy and efficiency of proposed algorithms.There are five chapters which are described as follows.In chapter 1,we introduce the background and development of our research,the main results of this thesis,and some Some relevant basic knowledge.In chapter 2,several new energy identities of metamaterial Maxwell's e-quations with the perfectly electric conducting(PEC)boundary condition are proposed and proved.These new energy identities are different from the Poynt-ing theorem.By using these new energy identities,it is proved that the Yee scheme on non-uniform rectangular meshes is stable in the discrete L2 and H1 norms when the Courant-Friedrichs-Lewy(CFL)condition is satisfied.Numeri-cal experiments on 2D and 3D are carried out and confirm our analysis,and the superconvergence in the discrete H1 norm is found.In chapter 3,we study the FDTD method for solving the Maxwell's equa-tions in a Cole-Cole dispersive medium.According to the features of Maxwell's equations and the numerical algorithm for fractional derivative term,we combine the L1 formula with the leap-frog FDTD method and the Crank-Nicolson FDTD method,then propose two efficient numerical schemes for Maxwell's equations in a Cole-Cole medium.We carry out the energy stability and error analysis rigorously by the energy method.Both schemes have been proved 2-? conver-gence rate in time and second convergence rate in space,respectively.Then,the numerical results on 2D and 3D are performed to confirm our theoretical analy-sis.In the process of using the L1 formula to calculate the fractional derivative,at each time layer requires all historical time layer information,which means that long time calculations and simulations will consume a lot of memory and CPU time.Thus,we propose an efficient FDTD algorithm based on the Sum-Of-Exponentials(SOE)approximation algorithm,and give detailed algorithm process.The 2D and 3D numerical results verify the accuracy and efficiency of our proposed algorithm.In chapter 4,we continue the research in Chapter 3 and propose a fully implicit FDTD method with second-order space-time accurate.This method is based on the Crank-Nicolson FDTD algorithm,which combines the L2-1? algo-rithm and the weighted approach.A rigorous analysis is carried out to show that the proposed scheme is unconditionally stable and has second-order accurate both in time and space.Numerical examples are presented to validate our theoretical findings.This newly proposed algorithm still needs all historical time information when calculating each time layer,which means that long time calculations and simulations will consume a lot of memory and CPU time.For this reason,on the bas is of the new algorithm,we use the SOE approximation idea to construct a new efficient FDTD algorithm,and give the detailed algorithm derivation process and implication details.Finally,through two-dimensional and three-dimensional numerical examples,we analyze and compare the two algorithms mentioned in this chapter.In chapter 5,we give the summary of this dissertation and the future research work prospects.