In this work, the ADI-FDTD method with fourth-order accuracy in time for the 2-D Maxwell's equations without sources and charges is proposed. We mainly focus on energy analysis of the proposed ADI-FDTD method. By using the energy method, we derive the numerical energy identity of the ADI-FDTD method and show that the ADI-FDTD method is approximately energy-preserving. In comparison with the energy in theory, the numerical one has two perturbation terms and can be used in computation in order to keep it approximately energy-preserving. Numerical experiments are given to show the performance of the proposed ADI-FDTD method which confirm the theoretical results.

In this work, an efficient scheme is proposed for the 2D damped sine-Gordon equation. The new scheme utilizes the unified weak Galerkin finite element method with or without stabilizer (UWG) in the spatial direction and the second-order explicit differentiation formula in the temporal direction. Both semi discrete schemes and fully discrete schemes are analyzed. First, we demonstrate the stability of the schemes. Subsequently, optimal error estimates are derived in the energy norm and the $L^2$ norm. Furthermore, by combining the proper orthogonal decomposition (POD) technique with the WG method, we develop a reduced-order algorithm, significantly improving computational efficiency. Finally, the theoretical results are validated by numerical experiments.