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A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport. (English) Zbl 1416.76147
Summary: 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.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
Software:
ECHO; Athena ; HLLE; RIEMANN
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