TY - JOUR T1 - Energy Identities and Stability Analysis of the Yee Scheme for 3D Maxwell Equations AU - Gao , Liping AU - Sang , Xiaorui AU - Shi , Rengang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 788 EP - 813 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0121 UR - https://global-sci.org/intro/article_detail/nmtma/15785.html KW - Finite difference time domain (FDTD) method, Maxwell equations, energy conservation, stability, convergence, CFL. AB -

In this paper numerical energy identities of the Yee scheme on uniform grids for  three dimensional Maxwell equations with periodic boundary conditions  are proposed and expressed in terms of the $L^2$, $H^1$ and $H^2$ norms. The relations between the $H^1$ or $H^2$ semi-norms and the magnitudes of the curls or the second curls of the fields in the Yee scheme are derived. By the $L^2$ form of the identity it is shown that the solution fields of the Yee scheme is approximately energy conserved. By the $H^1$ or $H^2$ semi norm of the identities, it is proved that the curls or the second curls of the solution of the Yee scheme are approximately magnitude (or energy)-conserved. From these numerical energy identities, the Courant-Friedrichs-Lewy (CFL) stability condition is re-derived, and the stability of the Yee scheme in the $L^2$,  $H^1$  and $H^2$ norms is then proved. Numerical experiments to compute the numerical energies and convergence orders in the  $L^2$, $H^1$ and $H^2$ norms are carried out and the computational results confirm the analysis of the Yee scheme on energy conservation and stability analysis.