In this paper, we develop and analyze two types of new energy-preserving local mesh-refined splitting finite difference time-domain (EP-LMR-S-FDTD) schemes for two-dimensional Maxwell's equations. For the local mesh refinements, it is challenging to define the suitable local interface schemes which can preserve energy and guarantee high accuracy. The important feature of the work is that based on energy analysis, we propose the efficient local interface schemes on the interfaces of coarse and fine grids that ensure the energy conservation property, keep spatial high accuracy and avoid oscillations and meanwhile, we propose a fast implementation of the EP-LMR-S-FDTD schemes, which overcomes the difficulty in solving unknowns on the “trifuecate structure” of refinement by first solving the values of coarse mesh unknowns and the average values of fine mesh unknowns on a line-structure and then solving the values of fine mesh unknowns and the coarse mesh unknown on an inverted “U-form” structure for each loop. The EP-LMR-S-FDTD schemes can be solved in a series of tridiagonal linear systems of unknowns which can be efficiently implemented at each time step. We prove the EP-LMR-S-FDTD schemes to be energy preserving and unconditionally stable. We further prove the convergence of the schemes and obtain the error estimates. Numerical experiments are given to show the performance of the EP-LMR-S-FDTD schemes which confirm theoretical results.

在本文中，我们针对二维Maxwell方程，开发并分析了两种新型的节能局部网格细化的有限差分时域（EP-LMR-S-FDTD）方案。对于局部网格细化，定义合适的局部接口方案以节省能量并确保高精度是一项挑战。这项工作的重要特点是，基于能量分析，我们在粗糙和精细网格的界面上提出了有效的局部界面方案，以确保节能特性，保持空间的高精度并避免振荡，同时，我们提出了一种快速实施方案EP-LMR-S-FDTD方案，通过首先求解线结构上的粗网格未知数的值和细网格未知数的平均值，然后求解细网格未知数和粗网格的值，克服了在精细化的“三级结构”上求解未知数的困难每个循环的倒置“ U形”结构上未知。可以在一系列未知的三对角线性系统中解决EP-LMR-S-FDTD方案，该方案可以在每个时间步均有效地实现。我们证明了EP-LMR-S-FDTD方案是节能且无条件稳定的。我们进一步证明了方案的收敛性，并获得了误差估计。数值实验表明了EP-LMR-S-FDTD方案的性能，证实了理论结果。