A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

TL;DR

This paper introduces a fourth-order accurate finite volume method for ideal MHD that maintains divergence-free magnetic fields using an innovative upwind constrained transport approach, improving accuracy and robustness over existing methods.

Contribution

The paper develops a novel fourth-order finite volume scheme for ideal MHD that integrates upwind constrained transport with PPM reconstruction, extending existing second-order methods.

Findings

Demonstrates formal fourth-order convergence for smooth problems

Shows robustness in discontinuous MHD problems

Achieves improved accuracy compared to second-order schemes

Abstract

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.