We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence errors. This leads us to design a new, arbitrary high-order numerical scheme for the induction equation in multiple space dimensions, based on a modification of the Spectral Difference (SD) method [1] with ADER time integration [2]. It appears as a natural extension of the Constrained Transport (CT) method. We show that it preserves $\mathrm{\nabla }\cdot \overrightarrow{B}=0$ exactly by construction, both in a local and a global sense. We compare our new method to the 3 RKDG variants and show that the magnetic energy evolution and the solution maps of our new SD-ADER scheme are qualitatively similar to the RKDG variant with divergence cleaning, but without the need for an additional equation and an extra variable to control the divergence errors.

本文针对带有感应方程的Runge-Kutta（RK）时间积分的高阶不连续Galerkin（DG）方法的三个变体进行了研究，分析了它们保留磁场无散度约束的能力。为了量化发散误差，我们使用基于表面项（用于测量整体发散误差）和体积项（用于测量局部发散误差）的范数。这导致我们基于对光谱差（SD）方法[1]进行了ADER时间积分[2]的修改，为多维空间中的感应方程设计了一种新的任意高阶数值方案。它似乎是“约束运输”（CT）方法的自然扩展。我们证明它可以保留$\mathrm{\nabla }\cdot \overrightarrow{乙}=0$完全是在本地和全球范围内构建而成。我们将我们的新方法与3个RKDG变体进行了比较，结果表明，新的SD-ADER方案的磁能演化和解图在质量上与采用散度清洗的RKDG变体相似，但是不需要额外的方程式和额外的变量以控制发散误差。