Symmetric Positive Semidefinite FDTD Subgridding Algorithms for Arbitrary Grid Ratios Without Compromising Accuracy

Instability has been a major problem in finite-difference time-domain (FDTD) subgridding methods. Reciprocity has been proposed to overcome the problem but with limited success in producing a symmetric positive semidefinite (SPD) system without compromising accuracy. In this paper, we algebraically derive both 2- and 3-D FDTD subgridding operators, which are SPD by construction, and independent of the grid ratio. Such operators have only nonnegative real eigenvalues, and hence the stability of the resulting explicit time marching is guaranteed. The 3-D operator, the algorithm of which is also applicable to 2-D analysis, further permits the use of a local time step, thus achieving a natural subgridding in both space and time. In addition, the interpretation of the proposed operators in the original FDTD formulation is also provided. Interestingly, not only interface unknowns but also subgrid unknowns are generated in a different way, as compared to the original FDTD, to simultaneously obtain an SPD system and to ensure accuracy. Extensive numerical simulations have demonstrated the accuracy, stability, and efficiency of the proposed new subgridding algorithms.