Explicit Matrix-Free Time-Domain Method in Unstructured Meshes and Its Application to Stable Simulation of General Unsymmetrical Systems

Abstract

The existing matrix-free time-domain (MFTD) method, though free of a matrix solution in arbitrary unstructured meshes, is not explicit in time marching, as a backward difference is utilized. The numerical system underlying the MFTD is unsymmetrical, and a traditional explicit simulation of such a system is absolutely unstable. In this article, we overcome this barrier and successfully develop a truly explicit MFTD method. In this method, a new explicit time marching scheme is created for simulating unsymmetrical systems, whose stability is theoretically proved and shown to be guaranteed. Meanwhile, the accuracy is not sacrificed; and the time step allowed by the traditional explicit method is not reduced to ensure the stability. As a result, we greatly improve the computational efficiency of the MFTD method without compromising its accuracy. In addition to the MFTD, the unsymmetrical systems are encountered in other numerical methods and analyses, such as the subgridding methods, the nonorthogonal FDTD methods, and the analysis of nonreciprocal problems. In this article, we show that the proposed new explicit method is a general method for stably simulating unsymmetrical systems. Hence, it can be utilized in other unsymmetrical methods to ensure the stability in an explicit time-domain simulation. The accuracy, efficiency, and stability of the proposed work have been demonstrated by extensive numerical experiments.