Energy-conserving Hermite methods for solving Maxwell's equations in dielectric and dispersive media are described and analyzed. In three space dimensions methods of order $2m$ to $2m+2$ require $(m+1)^3$ degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of $m$. We prove stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special seminorm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of electromagnetic wave propagation over thousands of wavelengths.

翻译：对介电介质和色散介质中求解麦克斯韦方程组的能量守恒厄米方法进行了描述与分析。在三维空间中，阶数为 $2m$ 至 $2m+2$ 的方法要求每个场变量在每个节点上具有 $(m+1)^3$ 个自由度，且可在与 $m$ 无关的时间步长下进行显式时间推进。我们证明了在仅受依赖域限制的时间步长下的稳定性，并给出了与插值过程相关的特殊半范数中的误差估计。数值实验表明，极高阶的厄米方法能够高效模拟数千个波长范围内的电磁波传播。