Critical analysis of a practical fourth order finite-difference time-domain algorithm for the solution of Maxwell's equations

Abstract

The finite-difference time-domain (FDTD) method is a highly effective numerical method of solvingMaxwell's equations in the time domain. Traditionally the approximation of the derivatives in Maxwell's equations is based on a central differencing scheme which is second order accurate (second order). The high complexity of today's electromagnetic problems necessitate a FDTD formulation that can use less computational memory and complete simulations faster than current second order FDTD formulations. Many researchers have studied the benefits of FDTD formulations based on fourth order approximations of the spatial derivatives (fourth order). However, none has presented a complete non-specialized case that leads to the simulation of practical antenna or electromagnetic problems. For this reason, and due to the complexity of the available fourth order formulations and the lack of comprehensive analysis of such FDTD formulations, none of the existing commercial electromagnetic software packages use any fourth order FDTD formulations for solving practical problems. The goal of this thesis is to implement, validate, and provide performance analysis of a practical FDTD scheme using fourth order accurate central differencing derivative approximations in space and second order accurate central differencing derivative approximations in time. The simplicity of the fourth order formulation presented in this thesis comes from that fact that it is derived from Taylor series expansions of a general function. The formulation of the FDTD updating equations is developed for general mediums as well as lumped circuit elements (voltage sources, resistors, capacitors, inductors, and diodes). Additionally, updating equations for fourth order convolutional perfectly matched layers (CPML) are derived. This formulation is straightforward, advantageous, and provides a practical fourth order FDTD formulation for electromagnetics applications. Verification and simulation accuracy of the developed fourth order formulation are confirmed through the application of Gaussian propagation, a cavity resonator, the radiation from a dipole antenna, antenna arrays, and the radar cross section calculation of a dielectric cube. Simulations of discontinuous boundaries are also explored in detail through the simulation of PEC objects and high permittivity objects. Various different methods of special fourth order updating equations are thoroughly tested at these boundaries and the results are analyzed. The computational advantages of the developed fourth order FDTD formulation are explored and results show reduced memory usage up to a factor of 6.97 and reduced simulation time up to a factor of 8.70 compared to the traditional second order FDTD formulation.