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A Constrained-Gradient Method to Control Divergence Errors in Numerical MHD

Hopkins, Philip F. (2016) A Constrained-Gradient Method to Control Divergence Errors in Numerical MHD. Monthly Notices of the Royal Astronomical Society, 462 (1). pp. 576-587. ISSN 0035-8711.

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In numerical magnetohydrodynamics (MHD), a major challenge is maintaining ∇⋅B=0. Constrained transport (CT) schemes achieve this but have been restricted to specific methods. For more general (meshless, moving-mesh, ALE) methods, ‘divergence-cleaning’ schemes reduce the ∇⋅B errors; however they can still be significant and can lead to systematic errors which converge away slowly. We propose a new constrained gradient (CG) scheme which augments these with a projection step, and can be applied to any numerical scheme with a reconstruction. This iteratively approximates the least-squares minimizing, globally divergence-free reconstruction of the fluid. Unlike ‘locally divergence free’ methods, this actually minimizes the numerically unstable ∇⋅B terms, without affecting the convergence order of the method. We implement this in the mesh-free code gizmo and compare various test problems. Compared to cleaning schemes, our CG method reduces the maximum ∇⋅B errors by ∼1–3 orders of magnitude (∼2–5 dex below typical errors if no ∇⋅B cleaning is used). By preventing large ∇⋅B at discontinuities, this eliminates systematic errors at jumps. Our CG results are comparable to CT methods; for practical purposes, the ∇⋅B errors are eliminated. The cost is modest, ∼30 per cent of the hydro algorithm, and the CG correction can be implemented in a range of numerical MHD methods. While for many problems, we find Dedner-type cleaning schemes are sufficient for good results, we identify a range of problems where using only Powell or ‘8-wave’ cleaning can produce order-of-magnitude errors.

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Hopkins, Philip F.0000-0003-3729-1684
Additional Information:© 2016 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society. Accepted 2016 June 29. Received 2016 May 24. In original form 2015 September 17. First published online July 4, 2016. We thank our anonymous referee for a number of helpful suggestions and additional tests. Support for PFH was provided by the Gordon and Betty Moore Foundation through Grant no. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics, an Alfred P. Sloan Research Fellowship, NASA ATP Grant NNX14AH35G, and NSF Collaborative Research Grant no. 1411920. Numerical calculations were run on the Caltech compute cluster ‘Zwicky’ (NSF MRI award #PHY-0960291) and allocation TG-AST130039 granted by the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the NSF.
Group:TAPIR, Moore Center for Theoretical Cosmology and Physics
Funding AgencyGrant Number
Gordon and Betty Moore Foundation776
Alfred P. Sloan FoundationUNSPECIFIED
Caltech Moore Center for Theoretical Cosmology and PhysicsUNSPECIFIED
Subject Keywords:hydrodynamics instabilities turbulence methods: numerical cosmology: theory
Issue or Number:1
Record Number:CaltechAUTHORS:20151022-135013652
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Official Citation:Philip F. Hopkins A constrained-gradient method to control divergence errors in numerical MHD MNRAS (October 11, 2016) Vol. 462 576-587 doi:10.1093/mnras/stw1578 first published online July 4, 2016
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:61432
Deposited By: Joy Painter
Deposited On:22 Oct 2015 21:14
Last Modified:03 Oct 2019 09:08

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