A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport - Felker, Kyle Gerard; Stone, James M. (Journal of computational physics, 2018)
•A fourth-order accurate method for the numerical solution of the equations of ideal magnetohydrodynamics (MHD) is proposed.•The piecewise parabolic method (PPM) in the upwind constrained transport framework satisfies the divergence-free constraint.•Modern PPM limiters provide MHD shock-capturing abilities while maintaining full fourth-order accuracy for smooth profiles.•Fourth-order accuracy yields significant improvement in solution errors relative to traditional second-order schemes. We present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.