In electrical engineering and physics, Maxwell's equations play a very important role and have a variety of applications. In this paper, we propose and analyse a novel class of time domain conservative schemes in rectangular coordinate to solve Maxwell's equations with periodic boundary conditions. This class of methods is explicit and easy to implement with low computational cost and memory storage. The error estimates presented in this paper demonstrate that the numerical solutions obtained by this class of conservative schemes can achieve arbitrarily high-order accuracy. The discrete divergences in numerical experiments motivate us to show that the conservative scheme is divergence-free by the rigorous numerical analysis, which is of great importance in the sense of geometric numerical integration. The discrete energy monitors give light to the statement that the conservative scheme can keep the linear relationship among some qualitative features of the underlying Maxwell's equations.

在电气工程和物理学中，麦克斯韦方程组扮演着非常重要的角色，有着多种应用。在本文中，我们提出并分析了一类新的时域保守方案直角坐标求解具有周期性边界条件的麦克斯韦方程组。这类方法是明确的，易于实现，计算成本和内存存储量低。本文给出的误差估计表明，通过此类保守方案获得的数值解可以达到任意高阶精度。数值实验中的离散发散促使我们通过严格的数值分析证明保守方案是无发散的，这在几何数值积分意义上非常重要。离散能量监视器阐明了保守方案可以保持基本麦克斯韦方程组的一些定性特征之间的线性关系的说法。