A Divergence-Free High-Order Spectral Difference Method with Constrained Transport for Ideal Compressible Magnetohydrodynamics

When the high-order Spectral Difference (SD) method is used to discretize ideal magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint del . B = 0 for the magnetic field over long time integration. To ensure that the discrete del . B equals to zero exactly and globally, the SD method is integrated with an unstaggered Constrained Transport approach (SDCT) by replacing the magnetic field with the curl of the magnetic potential at every time step. The SDCT method stores the variables for the hydrodynamics and the magnetic field at the same set of solution points, which avoids designing 2D Riemann solvers and preserves the compactness of the stencil for spatial discretization. Moreover, the additional computational cost is less than 1/8 of that without the constrained transport. Meanwhile, the SDCT method is found to have excellent convergence in test cases with and without shocks.