We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampere--Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.
翻译:对于Maxwell的方程式的初始边界值问题,我们引入了一种高阶样样样的几何方法。这种方法是几何方法,因为它在结构上分解了设计连续系统所涉及的两种不同形式的德拉姆序列。Ampere-Maxwell和Faraday等方程式必须坚固,使系统能够溶解,同时使两个离散的霍杰恒星操作员能够使用。通过利用从外部微积分中选择的样条空间和概念的特性,引入了一种非标准明确的时间配制,其依据是显示Kronecker产品结构的矩阵的线性系统的解决方案,而不是标准文献中的质量矩阵的解决方案。这些矩阵来自离散环境中的外部(红宝石)产品的应用,它们提供了与域的几何测量或材料参数无关的Kronecker产品结构。由此产生的计划保留了已知方法的可取的节能特性。新提出的办法的计算优势是通过复杂的分析和三个层面的数值实验加以研究的。