  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
$\nabla\cdot\vec{B}=0$ 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.
  [1] Liu Y., Vinokur M., Wang Z.J. (2006) Discontinuous Spectral Difference
Method for Conservation Laws on Unstructured Grids. In: Groth C., Zingg D.W.
(eds) Computational Fluid Dynamics 2004. Springer, Berlin, Heidelberg
  [2] Dumbser M., Castro M., Par\'es C., Toro E.F (2009) ADER schemes on
unstructured meshes for nonconservative hyperbolic systems: Applications to
geophysical flows. In: Computers & Fluids, Volume 38, Issue 9
