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3D Meshfree Magnetohydrodynamics

  • Stephan Rosswog
  • Daniel Price
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 65)

Abstract

We describe a new method to include magnetic fields into smooth particle hydrodynamics. The derivation of the self-gravitating hydrodynamics equations from a variational principle is discussed in some detail. The non-dissipative magnetic field evolution is instantiated by advecting so-called Euler potentials. This approach enforces the crucial ∇°B=0-constraint by construction. These recent developments are implemented in our three-dimensional, self-gravitating magnetohydrodynamics code MAGMA. A suite of tests is presented that demonstrates the superiority of this new approach in comparison to previous implementations.

Key words

astrophysics magnetohydrodynamics smoothed particle hydrodynamics magnetic fields Euler potentials 

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References

  1. 1.
    T. Alexander, Stellar processes near the massive black hole in the Galactic center, Phys. Rep., 419 (2005), pp. 65–142.CrossRefMathSciNetGoogle Scholar
  2. 2.
    H. Alfven, Cosmical Electrodynamics, Oxford University Press, Oxford, 1951.Google Scholar
  3. 3.
    D. S. Balsara, Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics, ApJS, 116 (1998), pp. 133−+.CrossRefGoogle Scholar
  4. 4.
    D. S. Balsara, Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics, J. Comp. Phys., 174 (2001), pp. 614–648.zbMATHCrossRefGoogle Scholar
  5. 5.
    A. A. Barmin, A. G. Kulikovskiy, and N. V. Pogorelov, Shock-Capturing Approach and Nonevolutionary Solutions in Magnetohydrodynamics, J. Comp. Phys., 126 (1996), pp. 77–90.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    W. Benz, Smooth particle hydrodynamics: A review, in Numerical Modeling of Stellar Pulsations, J. Buchler, ed., Kluwer Academic Publishers, Dordrecht, 1990, p. 269.Google Scholar
  7. 7.
    W. Benz, R. Bowers, A. Cameron, and W. Press, Dynamic mass exchange in doubly degenerate binaries. i — 0.9 and 1.2 solar mass stars, ApJ, 348 (1990), p. 647.CrossRefGoogle Scholar
  8. 8.
    T. Boyd and J. Sanderson The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.zbMATHGoogle Scholar
  9. 9.
    M. Brio and C. C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, Journal of Computational Physics, 75 (1988), pp. 400–422.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    L. Brookshaw, A method of calculating radiative heat diffusion in particle simulations, Proceedings of the Astronomical Society of Australia, 6 (1985), pp. 207–210.Google Scholar
  11. 11.
    S.-H. Cha and A. P. Whitworth, Implementations and tests of Godunov-type particle hydrodynamics, MNRAS, 340 (2003), pp. 73–90.CrossRefGoogle Scholar
  12. 12.
    J. E. Chow and J. Monaghan, Ultrarelativistic sph J. Computat. Phys., 134 (1997), p. 296.zbMATHCrossRefGoogle Scholar
  13. 13.
    T. E. Clarke, P. P. Kronberg, and H. Böhringer, A New Radio-X-Ray Probe of Galaxy Cluster Magnetic Fields, ApJL, 547 (2001), pp. L111–L114.CrossRefGoogle Scholar
  14. 14.
    R. B. Dahlburg and J. M. Picone, Evolution of the Orszag-Tang vortex system in a compressible medium. I—Initial average subsonic flow, Physics of Fluids B, 1 (1989), pp. 2153–2171.CrossRefMathSciNetGoogle Scholar
  15. 15.
    W. Dai and P. R. Woodward, Extension of the Piecewise Parabolic Method to Multidimensional Ideal Magnetohydrodynamics, J. Comp. Phys., 115 (1994), pp. 485–514.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    C. Eckart, Variation principles of hydrodynamics, Physics of Fluids, 3 (1960), p. 421.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. Euler, De curva hypergeometrica hac aequatione expressa y=…, Novi Commentarrii Acad. Sci. Petropolitanae, 14 (1769), p. 270.Google Scholar
  18. 18.
    T. A. Gardiner and J. M. Stone, An unsplit Godunov method for ideal MHD via constrained transport, J. Comp. Phys., 205 (2005), pp. 509–539.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R. A. Gingold and J. J. Monaghan, Smoothed particle hydrodynamics—Theory and application to non-spherical stars, MNRAS, 181 (1977), pp. 375–389.zbMATHGoogle Scholar
  20. 20.
    S. Gottloeber, G. Yepes, C. Wagner, and R. Sevilla The Mare Nostrum Universe, ArXiv Astrophysics e-prints, (2006).Google Scholar
  21. 21.
    A. Heger, S. E. Woosley, and H. C. Spruit Presupernova Evolution of Differentially Rotating Massive Stars Including Magnetic Fields ApJ, 626 (2005), pp. 350–363.CrossRefGoogle Scholar
  22. 22.
    D. Heggie and P. Hut, The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics, The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics, by Douglas Heggie and Piet Hut. Cambridge University Press, 2003, 372 pp., Feb. 2003.Google Scholar
  23. 23.
    L. Hernquist and N. Katz, Treesph—a unification of sph with the hierarchical tree method, ApJS, 70 (1989), p. 419.CrossRefGoogle Scholar
  24. 24.
    W. R. Hix, A. M. Khokhlov, J. C. Wheeler, and F.-K. Thielemann, The Quasi-Equilibrium-reduced alpha-Network, ApJ, 503 (1998), pp. 332−+.CrossRefGoogle Scholar
  25. 25.
    C. Ho, T. Huang, and S. Gao, Contributions to the high-degree multipoles of neptunes magnetic field, J. Geophys. Res., 102 (1997), p. 393.Google Scholar
  26. 26.
    S.-I. Inutsuka, Reformulation of Smoothed Particle Hydrodynamics with Riemann Solver, Journal of Computational Physics, 179 (2002), pp. 238–267.zbMATHCrossRefGoogle Scholar
  27. 27.
    J. Jackson Classical Electrodynamics, Wiley, New York, 3. ed., 1998.Google Scholar
  28. 28.
    P. Londrillo and L. Del Zanna, High-Order Upwind Schemes for Multidimensional Magnetohydrodynamics, ApJ, 530 (2000), pp. 508–524.CrossRefGoogle Scholar
  29. 29.
    L. Lucy, A numerical approach to the testing of the fission hypothesis, The Astronomical Journal, 82 (1977), p. 1013.CrossRefGoogle Scholar
  30. 30.
    M.-M. Mac Low and R. S. Klessen, Control of star formation by supersonic turbulence, Reviews of Modern Physics, 76 (2004), pp. 125–194.CrossRefGoogle Scholar
  31. 31.
    J. Monaghan and J. Lattanzio, A refined particle method for astrophysical problems, A&A, 149 (1985), p. 135.zbMATHGoogle Scholar
  32. 32.
    J. J. Monaghan, Smoothed particle hydrodynamics, Ann. Rev. Astron. Astrophys., 30 (1992), p. 543.CrossRefGoogle Scholar
  33. 33.
    J. J. Monaghan, SPH and Riemann Solvers, Journal of Computational Physics, 136 (1997), pp. 298–307.zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    J. J. Monaghan, SPH compressible turbulence, MNRAS, 335 (2002), pp. 843–852.CrossRefGoogle Scholar
  35. 35.
    J. J. Monaghan, Smoothed particle hydrodynamics, Reports of Progress in Physics, 68 (2005), pp. 1703–1759.CrossRefMathSciNetGoogle Scholar
  36. 36.
    J. J. Monaghan and D. J. Price, Variational principles for relativistic smoothed particle hydrodynamics, MNRAS, 328 (2001), pp. 381–392.CrossRefGoogle Scholar
  37. 37.
    J. Morris and J. Monaghan, A switch to reduce sph viscosity, J. Comp. Phys., 136 (1997), p. 41.zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    J. P. Morris, Analysis of smoothed particle hydrodynamics with applications, PhD thesis, Monash University, Melbourne, Australia, 1996.Google Scholar
  39. 39.
    S. Orszag and C. Tang, Small-scale structure in of two-dimensional magnetohydrodynamic turbulence, Journ. Fluid Mech., 90 (1979), p. 129.CrossRefGoogle Scholar
  40. 40.
    C. Peymirat and D. Fontaine, A numerical method to compute euler potentials, Ann. Geophysicae, 17 (1999), p. 328.CrossRefGoogle Scholar
  41. 41.
    J. M. Picone and R. B. Dahlburg, Evolution of the Orszag-Tang vortex system in a compressible medium. II—Supersonic flow, Physics of Fluids B, 3 (1991), pp. 29–44.CrossRefGoogle Scholar
  42. 42.
    D. Price, Magnetic Fields in Astrophysics, PhD thesis, University of Cambridge, arXiv:astro-ph/0507472, 2004.Google Scholar
  43. 43.
    D. Price and J. Monaghan, An energy-conserving formalism for adaptive gravitational force softening in sph and n-body codes, MNRAS, 374 (2007), p. 1347.CrossRefGoogle Scholar
  44. 44.
    D. Price and S. Rosswog, Producing ultra-strong magnetic fields in neutron star mergers, Science, 312 (2006), p. 719.CrossRefGoogle Scholar
  45. 45.
    D. J. Price, Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH, ArXiv e-prints, 709 (2007).Google Scholar
  46. 46.
    D. J. Price, splash: An Interactive Visualisation Tool for Smoothed Particle Hydrodynamics Simulations, Publications of the Astronomical Society of Australia, 24 (2007), pp. 159–173.CrossRefMathSciNetGoogle Scholar
  47. 47.
    D. J. Price and J. J. Monaghan, Smoothed Particle Magnetohydrodynamics —I. Algorithm and tests in one dimension, MNRAS, 348 (2004), pp. 123–138.CrossRefGoogle Scholar
  48. 48.
    D. J. Price, Smoothed Particle Magnetohydrodynamics—III. Multidimensional tests and the ∇. B=0 constraint, MNRAS, 364 (2005), pp. 384–406.Google Scholar
  49. 49.
    S. Rosswog, Last moments in the life of a compact binary, Rev. Mex. Astron. Astrophys., 27 (2007), pp. 57–79.Google Scholar
  50. 50.
    S. Rosswog and M. B. Davies, High-resolution calculations of merging neutron stars — I. Model description and hydrodynamic evolution, MNRAS, 334 (2002), pp. 481–497.CrossRefGoogle Scholar
  51. 51.
    S. Rosswog, M. B. Davies, F.-K. Thielemann, and T. Piran, Merging neutron stars: asymmetric systems, A&A, 360 (2000), pp. 171–184.Google Scholar
  52. 52.
    S. Rosswog, E. Ramirez-Ruiz, and R. Hix, Atypical thermonuclear supernovae from tidally crushed white dwarfs, ApJ, (2008).Google Scholar
  53. 53.
    S. Rosswog, E. Ramirez-Ruiz, and R. Hix, Simulating black hole white dwarf encounters, Comp. Phys. Comm, (2008).Google Scholar
  54. 54.
    S. Rosswog, E. Ramirez-Ruiz, and R. Hix, Tidal disruption and ignition of white dwarfs by intermediate-mass black holes, in prep., (2008).Google Scholar
  55. 55.
    S. Rosswog and M. Liebendörfer, High-resolution calculations of merging neutron stars — II. Neutrino emission, MNRAS, 342 (2003), pp. 673–689.CrossRefGoogle Scholar
  56. 56.
    S. Rosswog and D. Price, Magma: a magnetohydrodynamics code for merger applications, MNRAS, 379 (2007), pp. 915–931.CrossRefGoogle Scholar
  57. 57.
    G. Rüdiger and R. Hollerbach, The magnetic universe: geophysical and astrophysical dynamo theory, The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory, by Günther Rüdiger, Rainer Hollerbach, pp. 343. ISBN 3-527-40409-0. Wiley-VCH, August 2004., Aug. 2004.Google Scholar
  58. 58.
    D. Ryu and T. W. Jones, Numerical magetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow, ApJ, 442 (1995), pp. 228–258.CrossRefGoogle Scholar
  59. 59.
    H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Relativistic equation of state of nuclear matter for supernova and neutron star, Nuclear Physics, A 637 (1998), p. 435.CrossRefGoogle Scholar
  60. 60.
    H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Relativistic equation of state of nuclear matier for supernova explosion, Progress of Theoretical Physics, 100 (1998), pp. 1013–1031.CrossRefGoogle Scholar
  61. 61.
    G. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 43 (1978), pp. 1–31.CrossRefMathSciNetGoogle Scholar
  62. 62.
    R. Speith, Untersuchung von Smoothed Particle Hydrodynamics anhand astrophysikalischer Beispiele, PhD thesis, Eberhard-Karls-Universität Tübingen, 1998.Google Scholar
  63. 63.
    V. Springel, The cosmological simulation code GADGET-2, MNRAS, 364 (2005), pp. 1105–1134.CrossRefGoogle Scholar
  64. 64.
    V. Springel and L. Hernquist, Cosmological smoothed particle hydrodynamics simulations: the entropy equation, MNRAS, 333 (2002), pp. 649–664.CrossRefGoogle Scholar
  65. 65.
    D. Stern, Euler potentials, American Journal of Physics, 38 (1970), p. 494.CrossRefGoogle Scholar
  66. 66.
    D. P. Stern, The Motion of Magnetic Field Lines, Space Science Reviews, 6 (1966), p. 147.CrossRefGoogle Scholar
  67. 67.
    J. M. Stone, J. F. Hawley, C. R. Evans, and M. L. Norman, A test suite for magnetohydrodynamical simulations, ApJ, 388 (1992), pp. 415–437.CrossRefGoogle Scholar
  68. 68.
    C. Thompson and R. C. Duncan, Neutron star dynamos and the origins of pulsar magnetism, ApJ, 408 (1993), pp. 194–217.CrossRefGoogle Scholar
  69. 69.
    L. M. Widrow, Origin of galactic and extragalactic magnetic fields, Reviews of Modern Physics, 74 (2002), pp. 775–823.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stephan Rosswog
    • 1
  • Daniel Price
    • 2
  1. 1.Jacobs University BremenBremenGermany
  2. 2.School of PhysicsUniversity of ExeterExeterUK

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