Investigation of numerical time-integrations of Maxwell's equations using the staggered grid spatial discretization

摘要

The Yee-method is a simple and elegant way of solving the time-dependent Maxwell's equations. On the other hand, this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible time-steps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the time-step can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yee-method. Three other numerical methods are discussed: the Namiki-Zheng-Chen-Zhang alternating direction implicit method (NZCZ), the Kole-Figge-de Raedt method (KFR) and a Krylov-space method. All methods are discussed with non-homogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZ method without any computer algebraic tool. Moreover, we accelerate the Krylov-space method with a skew-symmetric formulation of the semi-discretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in ID numerical tests.