Abstract
The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.
Highlights
One of the most popular methods in computational electromagnetics is the finitedifference time-domain (FDTD) method [1,2]
The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method
The formulations from time-collocated to leapfrog fundamental schemes have been presented for alternating direction implicit (ADI) and CDI FDTD methods
Summary
One of the most popular methods in computational electromagnetics is the finitedifference time-domain (FDTD) method [1,2]. While satisfying Gauss’s law, the latter method exploits the efficient fundamental scheme and finds useful applications Both complying-divergence implicit (CDI) FDTD methods involve time-collocated electric and magnetic fields. In these implementations, the update equations comprise simple and concise RHS that contain only the intrinsic matrix operator partitions, but without their complicated products that exist in the conventional form. This can be seen from the implementation of (1) and (2) using their update equations in terms of matrix operator partitions (4) and (5) as
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