A leapfrog scheme for the unconditionally stable complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method is presented. The leapfrog CDI-FDTD method is formulated to comprise distinctive implicit and explicit update procedures. The implicit update procedures are in the fundamental form with operator-free right-hand sides (RHS), whereas the explicit ones are compatible and correspond to the familiar explicit FDTD method. Using the Fourier (von Neumann) approach, the stability of the leapfrog CDI-FDTD method is analyzed. The eigenvalues of amplification matrix are obtained to prove the unconditional stability. Furthermore, the leapfrog alternating direction implicit (ADI) FDTD and CDI-FDTD methods are discussed and compared, including the RHS floating-point operations (flops) count, for-loops and memory. The leapfrog CDI-FDTD requires merely half the flops of leapfrog ADI-FDTD at RHS while retaining the same left-hand sides (LHS) of implicit update equations and the same number of for-loops without much extra memory. Discussions and numerical results are presented to demonstrate the advantages of leapfrog CDI-FDTD method including unconditional stability, complying divergence, and efficient leapfrog scheme with reduced RHS flops.

中文翻译：

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顺散散度隐式时域有限差分法的蛙跳方案


提出了一种无条件稳定顺散隐式(CDI)时域有限差分(FDTD)方法的蛙跳方案。跨越式 CDI-FDTD 方法被制定为包括独特的隐式和显式更新过程。隐式更新过程采用无运算符右侧 (RHS) 的基本形式，而显式更新过程则兼容并对应于熟悉的显式 FDTD 方法。利用傅立叶（冯·诺依曼）方法，分析了蛙跳式CDI-FDTD方法的稳定性。得到放大矩阵的特征值，证明了其无条件稳定性。此外，还讨论并比较了蛙跳交替方向隐式 (ADI) FDTD 和 CDI-FDTD 方法，包括 RHS 浮点运算（触发器）计数、for 循环和内存。跨越式 CDI-FDTD 在 RHS 处仅需要跨越式 ADI-FDTD 的一半，同时保留相同的隐式更新方程左侧 (LHS) 和相同数量的 for 循环，而无需太多额外内存。讨论和数值结果证明了跨越式 CDI-FDTD 方法的优点，包括无条件稳定性、遵守散度以及具有减少 RHS flop 的高效跨越式方案。