Investigation of Numerical Time-Integrations of the Maxwell’s Equations Using the Staggered Grid Spatial Discretization

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-ChenZhang (NZCZ) ADI method, 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 in the approximation of the matrix exponential 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 1D numerical tests. Index Terms FDTD Method, Stability, Unconditional Stability