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The modification of the celebrated Yee scheme from Maxwell equations tomagnetohydrodynamics is often referred to as the constrained transportapproach. Constrained transport can be viewed as a sort of predictor-correctormethod for updating the magnetic field, where a magnetic field value is firstpredicted by a method that does not preserve the divergence-free condition onthe magnetic field, followed by a correction step that aims to control thesedivergence errors. This strategy has been successfully used in conjunction witha variety of shock-capturing methods including WENO, central, and wavepropagation schemes. In this work we show how to extend the basic CT frameworkto the discontinuous Galerkin finite element method on both 2D and 3D Cartesiangrids. We first review the entropy-stability theory for semi-discrete DGdiscretizations of ideal MHD, which rigorously establishes the need for amagnetic field that satisfies the following conditions: (1) the divergence ofthe magnetic field is zero on each element, and (2) the normal components ofthe magnetic field are continuous across element edges/faces. In order toachieve such a globally divergence-free magnetic field, we introduce a novel CTscheme that is based on two ingredients: (1) we introduce an element-centeredmagnetic vector potential that is updated via a discontinuous Galerkin schemeon the induction equation; and (2) we define a mapping that takeselement-centered magnetic field values and element-centered magnetic vectorpotential values and creates on each edge /face a representation of the normalcomponent of the magnetic field; this representation is then mapped back to theelements to create a globally divergence-free element-centered representationof the magnetic field. For problems with shock waves, we make use of so-calledmoment-based limiters to control oscillations in the conserved quantities.

High-Order Discontinuous Galerkin Finite Element Methods with Globally Divergence-Free Constrained Transport for Ideal MHD