Energy Identities and Stability Analysis of the Yee Scheme for 3D Maxwell Equations

Abstract

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.