Unconditionally Stable Divergence-Free Vector-Based Meshless Method for Transient Electromagnetic Analysis

A vector-based meshless method or vector meshless method, which is proven to be divergence-free, was recently proposed for transient electromagnetic analysis. In this method, an explicit finite-difference scheme is used for approximating the time derivatives in Maxwell’s equations. Therefore, the stability criterion constraints the maximum time-step. This paper presents the alternating-direction implicit (ADI) formulation of the vector meshless method based on a different form of the ADI algorithm named the leapfrog ADI scheme. The computational cost of the leapfrog ADI scheme is less than the conventional ADI algorithm and algebraically equivalent. In the proposed vector meshless method, Courant–Friedrichs–Lewy limit does not constrain the time-step due to its implicit formulation. Moreover, the stability condition of the conventional vector meshless method and the proposed ADI vector meshless method are investigated through their amplification matrices in this paper. The results show that the proposed method has two features: unconditional stability and divergence-free property. The first one is obtained due to the incorporation of the leapfrog ADI algorithm, and the second one is achieved by the use of vector basis functions. The divergence-free property of the proposed method can lead to obtaining more accurate solutions than the conventional meshless method. In addition, some numerical examples are presented to verify the properties and the efficiency of the proposed method.