Explicit Matrix-Free Time-Domain Method in Unstructured Meshes

Abstract

It has been shown when a numerical system obtained from discretizing Maxwell’s equations is unsymmetrical, a traditional explicit time marching is absolutely unstable because an unsymmetrical matrix can have complex-valued eigenvalues, and no time step can be found to make its explicit time marching stable. In this work, we overcome this barrier and successfully develop a matrix-free time-domain (MFTD) method which is truly explicit, requiring no matrix solution in arbitrary unstructured meshes. Although the system matrix of the MFTD is highly unsymmetrical, we develop a new explicit time marching scheme and theoretically prove that this scheme is guaranteed to be stable despite the unsymmetrical system matrix. Meanwhile, the accuracy of the time marching is not sacrificed; and the time step size allowed by a traditional explicit method is not reduced to ensure the stability of the new explicit scheme. As a result, we eliminate the need for a backward-difference-based implicit scheme in the MFTD method, and thereby the series expansion required for obtaining an explicit inverse of the system matrix, greatly improving the computational efficiency of the MFTD method without compromising its accuracy.Extensive numerical experiments on both unstructured triangular and tetrahedral meshes, and comparisons with the original MFTD method and analytical results have validated the accuracy, efficiency, and stability of the proposed new explicit MFTD method. The proposed new explicit method for simulating unsymmetrical systems can also be utilized in other methods whose underlying numerical systems are unsymmetrical to guarantee their stability in time domain.