A High-Order Spectral Difference Solver for 2D Ideal MHD Equations With Constrained Transport

Abstract

Abstract When using the discontinuous Galerkin or Spectral Difference (SD) method to discretize ideal magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint for the magnetic field over long-period time integration. To tackle this challenge, the SD method is integrated with an unstaggered Constrained Transport approach (SDCT). In addition to solving the two-dimensional ideal MHD equations, one more equation describing the transport of the magnetic potential is introduced. After each time step, the magnetic field will be updated by computing the curl of the magnetic potential. This strategy preserves ∇ · B = 0 exactly by construction in the discrete sense. Meanwhile, the additional computational cost is only approximately 20% more than that without the constrained transport. Moreover, the inclusion of the constrained transport does not obstruct the implementation of the artificial viscosity for shock capturing. Several well-known benchmark test cases are studied in this paper using the SDCT method. In the magnetic field loop advection test, the proposed SDCT avoids spurious growth of magnetic energy, and the numerical dissipation is shown to decrease when increasing the polynomial degree while maintaining the total degrees of freedom. In the propagation of Alfvén wave problem, the high-order accuracies of the SDCT method are verified. In the Orzag-Tang vortex problem, the predicted pressure distribution and density contours match well with those in the reference [1]. Meanwhile, a mesh convergence study shows that the SDCT method equipped with the artificial viscosity terms can produce converged results even in the vicinity of shocks.