Local Discontinuous Galerkin And Finite Difference Methods For Electromagnetic Wave Equations

Maxwell equations are a set of partial differential equations governing the mutual conversion and propagation of electric field and magnetic field,and can be changed into second order wave equations for electric field or magnetic field by substitution of variables,called electromagnetic wave equations.Investigation of the wave equations and their numerical methods is a kind of method to solve electromagnetic problems.In this thesis,from electromagnetic wave equations,new conservation related to H~1 semi norms of field functions is investigated and identities of conservation are derived.In recent years energy-conserved finite difference methods and discontinuous Galerkin methods become popular methods for solving Maxwell equations.Combining the new energy conservation with the two popular methods,finite difference methods and local discontinuous Galerkin(LDG)methods for electromagnetic wave equations are studied.Numerical identities for the Crank-Nicolson scheme are derive,error estimate is analyzed and numerical experiment to confirm the theoretical analysis are provided.In the development of the LDG methods for the wave equations,a new variable is introduced and q new LDG scheme for electromagnetic wave equations is proposed.Energy conservation of the scheme is proved and implementation of the LDG method in programming is given.By the new energy methods the error of the LDG method is analyzed and optimal estimate is obtained.The detailed contents of the thesis are listed as follows:The LDG schemes for the electromagnetic wave equations with the perfectly electric conducting(PEC)boundary conditions are proposed by introducing substitution of a function.It is proved that the LDG scheme is conserved.New conversion of the schemes is proved and error estimate is provided.It is shown that the LDG scheme is super convergent.Based on the semi-discrete scheme,the full-discrete scheme with frog jumping time discretization is considered and numerical experiment is carried out.Computational results confirm the theoretical analysis on the scheme.Conservation of the electromagnetic wave equations with periodic boundary conditions is investigated and identities in terms of H~1,H~2 and H~3 semi-norms are derived and analyzed.It is shown that the curls,the second and third curls of the fields are conserved with under their L~2 norms.In addition,the relations between the identities from the wave equations and Maxwell equations are explained.It is found that the identities derived from the wave equations are equivalent to those from Maxwell equations.By the new conservation,the Crank-Nicolson finite difference scheme is analyzed.Numerical identities of the scheme are derived and then it is proved that the scheme is conserved with respect the magnitudes of the first,second the third curls.The errors of the scheme under H~1,H~2 and H~3 norms are estimated and super convergence of the scheme is obtained.Numerical experiments are carried out and computational results demonstrated the analysis on new conservation and super convergence.