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

•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 ab...

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Bibliographic Details
Published in:Journal of computational physics Vol. 375; no. C; pp. 1365 - 1400
Main Authors: Felker, Kyle Gerard, Stone, James M.
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.12.2018
Elsevier Science Ltd
Elsevier
Subjects:
Astrophysics
Computational fluid dynamics
Computational physics
Conservation laws
Constrained transport
Divergence
Finite volume method
Fluid flow
Fluid mechanics
Formulations
High-order finite volume method
Magnetic fields
Magnetohydrodynamics
MATHEMATICS AND COMPUTING
Numerical analysis
Numerical methods
Robustness (mathematics)
Theory of constraints
Transport
High-order finite volume method
Magnetohydrodynamics
Constrained transport
Numerical methods
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:•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.
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USDOE Office of Science (SC)
FG02-97ER25308; AST-1715277
National Science Foundation (NSF)
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.08.025