Energy Conservation Of The Second Order Electromagnetic Wave Equation And Energy Conservation Analysis Of The Difference Method

Energy conservation is property of electromagnetic fields.Study of energy conservation for Maxwell equations and numerical methods is an important research task.In this thesis the energy conservation for the general form of Maxwell equations is extended to the second order wave equations and the finite difference methods of the wave equations.The contents of study are:(1)The new energy conservation for second order wave equations with PEC(perfectly electric conducting)boundary conditions is proposed and new energy identities in H^1,H^2and H^3 norms are derived.By the new energy methods,the Crank-Nicolson(CN)scheme of the one dimensional(1D)wave equations is analysed,and it is proved that this scheme with periodic boundary condition is energy conserved and second order convergent in terms of the discrete H^1,H^2 and H^3 norms.Numerical experiments confirm the analysis of the CN scheme.(2)New energy identities of the 1D Maxwell equations with PEC boundary conditions and periodic boundary conditions are derived and new energy conservation of the fields in H^1and H^2 norms is then proposed.By the new energy identities,the numerical energy identities of CN-FDTD scheme for the 1D Maxwell equations are derived and the energy conservation as well as convergence of this scheme are proved.Numerical experiments are carried out and computational results verify the analysis of CN-FDTD on energy conservation and convergence.(3)We study the error correction of the split finite difference method for the two-dimensional Maxwell equation.By adding perturbation terms a new method to decrease the error produced by splitting Maxwell equations is proposed,then two modified splitting FDTD method(MS-FDTD?,MS-FDTD?)for the 2D Maxwell equations are given by using the splitting FDTD scheme(S-FDTD?).The amplification factors and numerical dispersionrelations of the two schemes are derived by the Fourier method and the unconditional stability of the two schemes is then proved.The modified energy identities of two schemes(MS-FDTD?,MS-FDTD?)are derived and show that the two schemes proposed are approximately energy conserved.Numerical experiments to compute numerical dispersion errors and to solve a simple wave guide problem are carried out.Computational results confirm the numerical analysis of the two schemes and show that MS-FDTD? has less numerical dispersion errors than S-FDTD?.