Tridiagonal matrix algorithm

Abstract: The recently developed two-directional unconditionally stable single-field (US-SF) finite-difference time-domain (FDTD) method is generalized to a 3-D. The method is based on the application of the Crank–Nicolson scheme to only one of Maxwell curl equations which leads to an unconditionally stable finite-difference solution of the 3-D vector wave equation in the time domain. The method is designated as a single field because only the electric (or magnetic) field is updated in each time step. The implicit equations for each time step can be solved using a tridiagonal matrix algorithm without a need for matrix inversion. To achieve this, we introduced a new modified time-splitting scheme for the multilevel difference equations. Unlike the existing time-splitting method used to solve such equations which results in instability when dealing with inhomogeneous media, the presented method remains stable. As an important feature of the proposed US-SF-FDTD method, the updating of the three field components can be executed simultaneously (in parallel) by applying multithreading, thereby significantly reducing runtime. The unconditional stability of the proposed method is proved analytically. The accuracy and computational efficiency of the proposed method are demonstrated by providing numerical examples and by comparison to other FDTD methods and to the analytic solution when available.