This is a sequel to our earlier work in which we described an implicit leapfrog scheme in conjunction with a higher order mixed finite element discretization of a system of Maxwell's equations. In our earlier work, we focussed on providing the error analysis for both the semidiscretization of the Maxwell's equations using an implicit leapfrog scheme that we invented as well as providing the error analysis for the full discretization using this time domain scheme in conjunction with higher order mixed finite elements from finite element exterior calculus. In this work, we record our initial results with extending our implicit leapfrog scheme from being a discretization that is second order accurate in time to an arbitrary (even) order accurate in time method. Towards this end, we provide here the complete error analysis for the semidiscretization in time and full discretization of the Maxwell's equations for the fourth order scheme. We leave the completion of our efforts in providing all the necessary proofs for the general scheme to an immediate future update of this work.

翻译：本文是对我们前期工作的延续。前期工作中，我们描述了一种结合高阶混合有限元离散的隐式蛙跳格式，用于求解麦克斯韦方程组系统。我们重点给出了自创隐式蛙跳格式对麦克斯韦方程组进行半离散化的误差分析，以及结合有限元外微积分中的高阶混合有限元对该时域格式进行全离散化的误差分析。本文记录了我们初步的研究成果：将隐式蛙跳格式从时间二阶精度离散推广为任意（偶数）阶精度时间离散方法。为此，我们给出了四阶格式下麦克斯韦方程组时间半离散化和全离散化的完整误差分析。关于更一般格式所需全部证明的完善工作，我们将在本文的后续更新中尽快完成。