Symmetric Splitting Exponential FDTD Scheme For The Maxwell’s Equations

In this thesis two kinds of schemes: the symmetric splitting exponential FDTDscheme(denoted by SSE-FDTD), and the high-order symmetric splitting exponentialFDTD scheme(denoted by HO-SSE-FDTD) for the two dimensional Maxwell’s equa-tions in a lossy medium are proposed by combined the symmetric energy-conservedsplitting FDTD scheme and the high order finite diference methods with exponentialFDTD methods. The energy conservation, stability, error estimate and convergence ofthe SSE-FDTD scheme for the2D Maxwell equations in a lossy medium in a rectangulardomain covered by perfectly electric conducting(PEC) boundary conditions are analyzedby new energy methods. It is proved that the SSE-FDTD scheme is energy conserved,unconditionally stable and convergent in the discrete L2norm and the discrete H1norm,respectively. Numerical experiments are carried out and confirm the theoretical analysis.The truncation error for the high-order symmetric splitting exponential FDTD schemeis derived and and computational method of this scheme is studied.The thesis is divided into four chapters.In the first Chapter, we introduce the background and importance of selected re-search project, including the Maxwell equations in a lossy medium and some numericalmethods of these equations.In the second Chapter, combined the symmetric energy-conserved splitting FDTDmethod with the exponential FDTD method, a new scheme called the symmetric splittingexponential FDTD scheme(SSE-FDTD) is proposed. The energy identity of SSE-FDTDin the discrete L2norm is derived and prove that this scheme is energy conserved and unconditionally stable. By deriving the truncation error of this new scheme,it is shownthat SSE-FDTD is second order accurate in both time and space, and proved that theSSE-FDTD scheme is second order convergent in the discrete L2norm. Numerical exper-iments are given and computational results conform the theoretical analysis on energyconservation, stability and convergence.In the third Chapter, the energy identities of SSE-FDTD in the discrete H1normare derived. By these identities it is proved that the SSE-FDTD scheme is uncondition-ally stable and second order convergent in the discrete H1norm. It is also shown thatthe SSE-FDTD scheme is super convergent in the discrete H1semi-norm. Experimentsare provided and confirm our theoretical analysis.In the forth Chapter, combining high-order diference methods with respect to spacevariables, the SSE-FDTD scheme and the exponential diference method, the high-ordersymmetric splitting exponential FDTD scheme of the two dimensional Maxwell’s equa-tions in a lossy medium is proposed. The systems of equations from the HO-SSE-FDTDscheme is derived and solving methods are studied. It is shown that the HO-SSE-FDTDare second order accurate in time and forth order accurate in space.