Research On Low Numerical Dispersion Finite Difference Time Domain Algorithm

The finite difference time domain(FDTD)method,as a typical time-domain numer-ical method in computational electromagnetic,starts directly from Maxwell's equations without performing complex matrix inversion operations.It is simple,intuitive and easy to master.Hence,it has been widely used in electromagnetic simulation.However,the core point of the FDTD algorithm is to use the finite difference scheme to discretize Maxwell's curl equations.Therefore,numerical dispersion errors will occur during the discrete pro-cess,which directly affects the calculation accuracy of the FDTD algorithm.Reducing the numerical dispersion error of the FDTD algorithm can not only improve its calculation ac-curacy and computation efficiency,but also can save computational time and memory.This dissertation mainly studies the FDTD algorithm with low numerical dispersion.After analyzing and summarizing several typical FDTD algorithms,combining and im-proving some existing optimization techniques,a variety of FDTD algorithms with low numerical dispersion are proposed,which greatly reduces the numerical dispersion error and improves accuracy and efficiency of the original algorithms.The main research con-tents of this paper are as follows:Firstly,the FDTD algorithm based on hexagonal grid with low numerical dispersion is studied.This dissertation introduces scaling factor to correct the medium parameters for the traditional H-FDTD method and a low numerical dispersion CH-FDTD algorithm is proposed.Compared with the traditional H-FDTD algorithm,the proposed CH-FDTD al-gorithm can achieve almost analytical solution,significantly reduce numerical dispersion,and greatly improve the accuracy of the algorithm.Secondly,this dissertation studied four typical unconditionally stable FDTD algo-rithms,and corresponding low numerical dispersion algorithms are proposed.The tradi-tional unconditionally stable split-step(SS)FDTD algorithm and the alternate direction implicit(ADI)FDTD algorithm are optimized by introducing the isotropic dispersion(ID)finite difference scheme into the traditional SS-FDTD algorithm and ADI-FDTD algo-rithm.The weighting factors and scaling factors for the SS-FDTD algorithm and ADI-FDTD algorithm in the ID finite difference scheme were re-derived,thence low numerical dispersion SS-FDTD algorithm and ADI-FDTD algorithm are proposed.This two kinds of unconditionally stable algorithms with low numerical dispersion can achieve almost analytic numerical phase velocity,and numerical dispersion error has been reduced by several orders of magnitude.Then,on the basis of the traditional locally one dimensional(LOD)FDTD algorithm for lossy media,the ID scheme for lossy media LOD-FDTD al-gorithm is introduced,and the LOD-FDTD algorithm for lossy media with low numerical dispersion is proposed.The weighting factor and two scale factors of the ID scheme of the LOD-FDTD algorithm for lossy media are re-derived.The deduced scale factors are used to modify the permittivity ?,the permeability ? and the conductivity ? of the loss medium,which greatly reduces the numerical phase error and numerical attenuation er-ror of the traditional lossy medium LOD-FDTD algorithm.Based on an unconditionally stable weighted laguerre polynomia(WLP)FDTD algorithm which have induced the ID scheme,a scaling factor is used to modify the dielectric constant and conductivity in the proposed optimized method,thereby significantly reducing the numerical dispersion error of the ID-WLP-FDTD algorithm.Then,the weakly conditionally stable FDTD algorithm with low numerical disper-sion is studied.This dissertation introduces optimized 3-D ID schemes into the 3-D hybrid implicit explicit(HIE)FDTD algorithm and proposed two optimized three-dimensional ID-HIE-FDTD algorithms.The weighting factors and scaling factors for the HIE-FDTD algorithm in the 3-D case are reformulated.The newly proposed optimization algorithms can not only greatly reduce the numerical dispersion of the original HIE-FDTD algorithm,achieve the almost analytical phase velocity,but also can obtain weaker stability condi-tions,thereby the proposed methods can meet the needs of accuracy and efficiency simul-taneously.Finally,the low numerical dispersion FDTD algorithm in isotropic and anisotropic dispersion media are studied respectively.Based on the typical auxiliary differential equa-tion(ADE)FDTD method in non-magnetized plasma,a non-magnetized plasma ADE-FDTD algorithm with low numerical dispersion is proposed.The numerical dispersion equation of the traditional ADE-FDTD algorithm is derived,and on this basis,the least squares fitting(LSF)technique is used to introduce two optimization factors and more sampling points,which can effectively reduce the numerical dispersion error and numeri-cal dissipation error of the original algorithm.Then,by introducing an optimization coeffi-cient into typical piecewise linear recursive convolution(PLRC)FDTD algorithm in mag-netized plasma,a low numerical dispersion magnetized plasma PLRC-FDTD algorithm is proposed.Numerical dispersion analysis of the optimized finite difference scheme val-idates its advantages,and numerical examples of electromagnetic pulse wave propagating in magnetized plasma slab demonstrate the high efficiency of the new proposed method.This dissertation had studied the low numerical dispersion FDTD algorithm system-atically and provided many kinds of FDTD algorithms with low numerical dispersion,which could laid a solid foundation for the high efficiency and wider application of the FDTD algorithm.