Abstract

This thesis proposes several new finite-difference time-domain (FDTD) methods to overcome shortcomings of current FDTD schemes: the new explicit methods have better numerical accuracy and the new implicit methods have unconditional stability; an error quantification method is described to evaluate the discretization error of a FDTD method; and a new concept of numerical loss in lossy materials is discussed, which has been neglected by the FDTD community. The new explicit methods are derived by optimizing the numerical dispersion relation. The 24-stencil method and the neighborhood-average method can have high accuracy in a given angular sector; or have zero anisotropy in the 2D and 3D cases. Combining the two methods, the neighborhood-average-24 method provides one order-of-magnitude lower accumulated phase error than other published methods, and can use as large as the Courant time step size. The correct numerical dispersion relations for the implicit alternating-direction-implicit (ADI) method are derived and verified with good agreement with the numerical experiments. The inconsistency in the literature concerning the dispersion relation is removed. Based on the high-accuracy, fully implicit and inefficient Crank-Nicolson scheme, several new efficient implicit methods are proposed, which have much smaller anisotropy and smaller discretization error than ADI. The numerical dispersion relations and the perturbation errors to the Crank-Nicolson scheme are given. It is shown that all the unconditionally-stable methods have their own time-step-size upper bounds to avoid non-physical attenuation, and have intrinsic spatial dispersion and intrinsic temporal dispersion. A method to quantify the discretization error of an FDTD scheme is developed and is used to compare the errors of various schemes. In lossy media, the relations between numerical phase and loss constants are derived for Yee's FDTD, ADI and the Crank-Nicolson-based methods, and verified with good agreement with numerical experiments. The numerical loss constant is always larger than its physical value, which implies that the electric field strengths computed by the FDTD methods in lossy media are smaller than the actual physical values. The numerical velocity in lossy media can be smaller or larger than its physical value. The finite-difference operators and the efficient splitting scheme proposed in the thesis are powerful tools in developing new FDTD methods.