Study Of Several High Order Numerical Methods For Electromagnetic Problem

In this thesis,we focus on finite difference(FD)methods with high order in time for second order electromagnetic wave equations and first order Maxwell equations.Based on the two kinds of equations,several FD methods with high order(or any high order)in time and second order or fourth order in space for wave guide problems and propagation problems in unbounded domains are proposed and studied on stability,and numerical dispersion.How to cope with or discretize the Berenger split-field perfectly matched layers(PML)is investigated and FD method with high order in time for PML is given.The merits or advantages of the new proposed FD methods for the two kinds of differential equations over the current FD methods are: explicit and simple in computation,higher or any high order in time and second order or fourth order in space,and good stability.Numerical experiments are carried out and computational results confirm the efficiency of the new FD methods.The details of the research are:(1)Two FD methods with high-order in time and second as well as fourth order in space for the second-order wave equation with the perfectly electric conducting(PEC)boundary conditions are proposed.The numerical dispersion relations and amplification factors of the new methods are derived,and the stability of the methods are then proved.It is shown that the stability of the two kinds of FD methods are not restricted by the ratio of time and spatial step sizes or the Courant-Friendrich-lewy(CFL)number.In numerical experiments the modules of amplification factors,numerical dispersion errors,absolute errors and convergence orders for the new FD methods are computed and compared with those values for the corresponding Crank-Nicolson(CN)schemes of the second order wave equations.It is shown that the maximal errors of the solutions of the new FD methods are less than those of the CN schemes.(2)The FD method with higher order in time and fourth order in space for the 2-D Maxwell’s equations with PEC boundary conditions is proposed.By the Fourier method it is proved that the new FD method is stable with no restriction on the ratio of time and spatial step sizes(unconditionally stable).Numerical experiments confirm the new FD method on stability and convergence order.In comparison with the Yee scheme.,computational results show that the new FD method has less errors than the Yee scheme.(3)Mur absorbing boundary conditions and Berenger PML are studied and the FD methods with higher order in time for the Berenger PML is proposed.A kind of wave propagation problem in an unbounded domain in a plane are solved by using new FD methods with the first order Mur absorb boundary condition and the Berenger PML.Computational results are compared with the Yee schemes and confirm the efficiency of new FD methods and the consistency with the commonly used absorbing boundary conditions.