A High-Order Unstaggered Constrained-Transport Method for the Three-Dimensional 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, Rossmanith, and Taetz [J. Comput. Phys., 230 (2011), pp. 3803--3829] the authors presented an unstaggered CT 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 Helzel, Rossmanith, and Taetz [J. Comput. Phys., 230 (2011), pp. 3803--3829] 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 nonconservative finite volume method for the magnetic vector potential equation; (2) we increase the spatial and temporal order of accuracy of the entire method to third order; and (3) we develop the method so that it is applicable on both Cartesian and logically rectangular mapped grids. The method-of-lines approach that is used in this work is based on a third-order accurate finite volume discretization in space coupled to a third-order strong stability preserving Runge--Kutta time-stepping method. The evolution equation for the magnetic vector potential is solved using a nonconservative finite volume method based on the approach of Castro et al. [Math. Comput., 79 (2010), pp. 1427--1472]. The curl of the magnetic potential is computed via a third-order accurate discrete operator that is derived from appropriate application of the divergence theorem and subsequent numerical quadrature on element faces. Special artificial resistivity limiters are used to control unphysical oscillations in the magnetic potential and field components across shocks. Several test computations are shown that confirm third-order accuracy for smooth test problems and high resolution for test problems with shock waves.