A HIGH-ORDER UNSTAGGERED CONSTRAINED TRANSPORT METHOD FOR THE 3D IDEAL MAGNETOHYDRODYNAMIC EQUATIONS BASED ON THE METHOD OF LINES

Abstract. Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must confront the challenge of controlling errors in the discrete divergence of the magnetic field. One approach that has been shown successful in stabilizing MHD calculations are constrained transport (CT) schemes. CT schemes can be viewed as predictor-corrector methods for updating the magnetic field, where a magnetic field value is first predicted by a method that does not exactly preserve the divergence-free condition on the magnetic field, followed by a correction step that aims to control these divergence errors. In Helzel et al. [J. Comp. Phys. 227, 9527 (2011)] the authors presented an unstaggered constrained transport method for the MHD equations on three-dimensional Cartesian grids. In this approach an evolution equation for the magnetic potential is solved during each time step and a divergence-free update of the magnetic field is computed by taking the curl of the magnetic potential. The evolution equation for the vector potential is only weakly hyperbolic, which requires special numerical treatment. A key step in this method is the use of dimensional splitting in order to overcome these difficulties. In this work we generalize the method of [J. Comp. Phys. 227, 9527 (2011)] in three important ways: (1) we remove the need for operator splitting by switching to an appropriate method of lines discretization and coupling this with a non-conservative finite volume method for the magnetic vector potential equation, (2) w