Abstract
In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L2 and H1 norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.
Highlights
The alternative direction implicit finite difference time domain (ADI-FDTD) methods, proposed in [1] [2], are interesting and efficient methods for numerical solutions of Maxwell equations in time domain, and cause many researchers’ work since ADI-FDTD overcomes the stability constraint of the FDTD scheme [3]. It was proved by Fourier methods in [4]-[8] that the ADI-FDTD methods are unconditionally stable and have reasonable numerical dispersion error; Reference [9] studied the divergence property; Reference [10] studied ADI-FDTD in a perfectly matched medium; Reference [11] gave an efficient PML implementation for the ADI-FDTD method
We focus our attention on structure with periodic boundary conditions and propose energy identities in L2 and H1 norms of the 2D Maxwell equations with periodic boundary conditions
Several modified energy identities of 2D-ADI-FDTD in terms of the discrete L2 and H1 norms are presented. By these identities it is proved that 2D-ADI-FDTD with the periodic boundary conditions is unconditionally stable and approximately energy conserved under the discrete L2 and H1 norms
Summary
The alternative direction implicit finite difference time domain (ADI-FDTD) methods, proposed in [1] [2], are interesting and efficient methods for numerical solutions of Maxwell equations in time domain, and cause many researchers’ work since ADI-FDTD overcomes the stability constraint of the FDTD scheme [3]. T. Yang tions and proved that ADI-FDTD is approximately energy conserved and unconditionally in the discrete L2 and H1 norms. We derive the energy identities of ADI-FDTD for the 2D Maxwell equations (2D-ADI-FDTD) with periodic boundary conditions by a new energy method. Several modified energy identities of 2D-ADI-FDTD in terms of the discrete L2 and H1 norms are presented. By these identities it is proved that 2D-ADI-FDTD with the periodic boundary conditions is unconditionally stable and approximately energy conserved under the discrete L2 and H1 norms.
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