An Arbitrary Lagrangian-Eulerian Discretization of MHD on 3D Unstructured Grids

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We present an arbitrary Lagrangian-Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured hexahedral grids. The method is formulated using an operator-split approach with three distinct phases: electromagnetic diffusion, Lagrangian motion, and Eulerian advection. The resistive magnetic dynamo equation is discretized using a compatible mixed finite element method with a 2nd order accurate implicit time differencing scheme which preserves the divergence-free nature of the magnetic field. At each discrete time step, electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. By virtue of the ... continued below

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Rieben, R N; White, D A; Wallin, B K & Solberg, J M June 12, 2006.

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We present an arbitrary Lagrangian-Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured hexahedral grids. The method is formulated using an operator-split approach with three distinct phases: electromagnetic diffusion, Lagrangian motion, and Eulerian advection. The resistive magnetic dynamo equation is discretized using a compatible mixed finite element method with a 2nd order accurate implicit time differencing scheme which preserves the divergence-free nature of the magnetic field. At each discrete time step, electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. By virtue of the compatible discretization method used, the invariants of Lagrangian MHD motion are preserved in a discrete sense. When the Lagrangian motion of the mesh causes significant distortion, that distortion is corrected with a relaxation of the mesh, followed by a 2nd order monotonic remap of the electromagnetic state variables. The remap is equivalent to Eulerian advection of the magnetic flux density with a fictitious mesh relaxation velocity. The magnetic advection is performed using a novel variant of constrained transport (CT) that is valid for unstructured hexahedral grids with arbitrary mesh velocities. The advection method maintains the divergence free nature of the magnetic field and is second order accurate in regions where the solution is sufficiently smooth. For regions in which the magnetic field is discontinuous (e.g. MHD shocks) the method is limited using a novel variant of algebraic flux correction (AFC) which is local extremum diminishing (LED) and divergence preserving. Finally, we verify each stage of the discretization via a set of numerical experiments.

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PDF-file: 44 pages; size: 5.6 Mbytes

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  • Journal Name: Journal of Computational Physics, vol. 226, no. 1, May 21, 2007, pp. 534 - 570; Journal Volume: 226; Journal Issue: 1

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  • Report No.: UCRL-JRNL-222113
  • Grant Number: W-7405-ENG-48
  • Office of Scientific & Technical Information Report Number: 936675
  • Archival Resource Key: ark:/67531/metadc902863

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  • June 12, 2006

Added to The UNT Digital Library

  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 22, 2016, 7:44 p.m.

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Rieben, R N; White, D A; Wallin, B K & Solberg, J M. An Arbitrary Lagrangian-Eulerian Discretization of MHD on 3D Unstructured Grids, article, June 12, 2006; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc902863/: accessed October 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.