A Leapfrog ADI-FDTD Method For TE Wave Maxwell-Debye Equations

Maxwell’s equations are a set of partial differential equations that accurately describe the relationship between electric and magnetic fields,charge density and current density.The permittivity or permeability in the constitutive relation of the dispersive medium depends on the frequency of the field.As a common dispersion medium model,Debye model is often used to describe the dispersion characteristics of soil,water,human tissue and other media.The leapfrog ADI-FDTD method is derived from the ADI-FDTD method.Its main idea is to divide a time step into two sub-time steps.The values of electricity field and magnetic field are taken in integer and semi-integer time steps at the same time.Then subtract a full time step from the second sub-time step,and combine it with the first sub-time step.On the basis of guaranteeing the unconditional stability of the format,the calculation of intermediate time step is reduced and the calculation efficiency is improved.In this thesis,we consider Maxwell’s equations in differential form,which describe the field at each point in the continuou s medium.The propagation of electromagnetic waves in complex dispersive media is controlled by the coupling of Maxwell’s equations and the equations describing macroscopic polarization.The Maxwell-Debye model is used to simulate the dynamic evolution of the macroscopic polarization driven by the electric field by appending to Maxwell’s equations a set of ordinary differential equations(ODE)by auxiliary differential equation(ADE)method.In this thesis,the leapfrog ADI-FDTD method is used to solve the MaxwellDebye model.Consider the periodic boundary conditions,and use the Yee scheme for discretization.The corresponding fully discretization scheme of the model is given.At the same time,we give the rigorous theoretical analysis of the stability and convergence.It is obtained that the scheme has a second-order optimal error estimate in time and space.Finally,a numerical example is used to verify thar the experimental results are consistent with the theoretical analysis results.