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Meshfree Methods for Partial Differential Equations IV pp 247–275Cite as

3D Meshfree Magnetohydrodynamics

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Part of the book series: Lecture Notes in Computational Science and Engineering ((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.

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References

  1. T. Alexander, Stellar processes near the massive black hole in the Galactic center, Phys. Rep., 419 (2005), pp. 65–142.

    Article  MathSciNet  Google Scholar 

  2. H. Alfven, Cosmical Electrodynamics, Oxford University Press, Oxford, 1951.

    Google Scholar 

  3. D. S. Balsara, Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics, ApJS, 116 (1998), pp. 133−+.

    Article  Google Scholar 

  4. D. S. Balsara, Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics, J. Comp. Phys., 174 (2001), pp. 614–648.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  Google Scholar 

  8. T. Boyd and J. Sanderson The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. S.-H. Cha and A. P. Whitworth, Implementations and tests of Godunov-type particle hydrodynamics, MNRAS, 340 (2003), pp. 73–90.

    Article  Google Scholar 

  12. J. E. Chow and J. Monaghan, Ultrarelativistic sph J. Computat. Phys., 134 (1997), p. 296.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Eckart, Variation principles of hydrodynamics, Physics of Fluids, 3 (1960), p. 421.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Euler, De curva hypergeometrica hac aequatione expressa y=…, Novi Commentarrii Acad. Sci. Petropolitanae, 14 (1769), p. 270.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. A. Gingold and J. J. Monaghan, Smoothed particle hydrodynamics—Theory and application to non-spherical stars, MNRAS, 181 (1977), pp. 375–389.

    MATH  Google Scholar 

  20. S. Gottloeber, G. Yepes, C. Wagner, and R. Sevilla The Mare Nostrum Universe, ArXiv Astrophysics e-prints, (2006).

    Google Scholar 

  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.

    Article  Google Scholar 

  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. L. Hernquist and N. Katz, Treesph—a unification of sph with the hierarchical tree method, ApJS, 70 (1989), p. 419.

    Article  Google Scholar 

  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−+.

    Article  Google Scholar 

  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. S.-I. Inutsuka, Reformulation of Smoothed Particle Hydrodynamics with Riemann Solver, Journal of Computational Physics, 179 (2002), pp. 238–267.

    Article  MATH  Google Scholar 

  27. J. Jackson Classical Electrodynamics, Wiley, New York, 3. ed., 1998.

    Google Scholar 

  28. P. Londrillo and L. Del Zanna, High-Order Upwind Schemes for Multidimensional Magnetohydrodynamics, ApJ, 530 (2000), pp. 508–524.

    Article  Google Scholar 

  29. L. Lucy, A numerical approach to the testing of the fission hypothesis, The Astronomical Journal, 82 (1977), p. 1013.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  31. J. Monaghan and J. Lattanzio, A refined particle method for astrophysical problems, A&A, 149 (1985), p. 135.

    MATH  Google Scholar 

  32. J. J. Monaghan, Smoothed particle hydrodynamics, Ann. Rev. Astron. Astrophys., 30 (1992), p. 543.

    Article  Google Scholar 

  33. J. J. Monaghan, SPH and Riemann Solvers, Journal of Computational Physics, 136 (1997), pp. 298–307.

    Article  MATH  MathSciNet  Google Scholar 

  34. J. J. Monaghan, SPH compressible turbulence, MNRAS, 335 (2002), pp. 843–852.

    Article  Google Scholar 

  35. J. J. Monaghan, Smoothed particle hydrodynamics, Reports of Progress in Physics, 68 (2005), pp. 1703–1759.

    Article  MathSciNet  Google Scholar 

  36. J. J. Monaghan and D. J. Price, Variational principles for relativistic smoothed particle hydrodynamics, MNRAS, 328 (2001), pp. 381–392.

    Article  Google Scholar 

  37. J. Morris and J. Monaghan, A switch to reduce sph viscosity, J. Comp. Phys., 136 (1997), p. 41.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. P. Morris, Analysis of smoothed particle hydrodynamics with applications, PhD thesis, Monash University, Melbourne, Australia, 1996.

    Google Scholar 

  39. S. Orszag and C. Tang, Small-scale structure in of two-dimensional magnetohydrodynamic turbulence, Journ. Fluid Mech., 90 (1979), p. 129.

    Article  Google Scholar 

  40. C. Peymirat and D. Fontaine, A numerical method to compute euler potentials, Ann. Geophysicae, 17 (1999), p. 328.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  42. D. Price, Magnetic Fields in Astrophysics, PhD thesis, University of Cambridge, arXiv:astro-ph/0507472, 2004.

    Google Scholar 

  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.

    Article  Google Scholar 

  44. D. Price and S. Rosswog, Producing ultra-strong magnetic fields in neutron star mergers, Science, 312 (2006), p. 719.

    Article  Google Scholar 

  45. D. J. Price, Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH, ArXiv e-prints, 709 (2007).

    Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  47. D. J. Price and J. J. Monaghan, Smoothed Particle Magnetohydrodynamics —I. Algorithm and tests in one dimension, MNRAS, 348 (2004), pp. 123–138.

    Article  Google Scholar 

  48. D. J. Price, Smoothed Particle Magnetohydrodynamics—III. Multidimensional tests and the ∇. B=0 constraint, MNRAS, 364 (2005), pp. 384–406.

    Google Scholar 

  49. S. Rosswog, Last moments in the life of a compact binary, Rev. Mex. Astron. Astrophys., 27 (2007), pp. 57–79.

    Google Scholar 

  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.

    Article  Google Scholar 

  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. S. Rosswog, E. Ramirez-Ruiz, and R. Hix, Atypical thermonuclear supernovae from tidally crushed white dwarfs, ApJ, (2008).

    Google Scholar 

  53. S. Rosswog, E. Ramirez-Ruiz, and R. Hix, Simulating black hole white dwarf encounters, Comp. Phys. Comm, (2008).

    Google Scholar 

  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. S. Rosswog and M. Liebendörfer, High-resolution calculations of merging neutron stars — II. Neutrino emission, MNRAS, 342 (2003), pp. 673–689.

    Article  Google Scholar 

  56. S. Rosswog and D. Price, Magma: a magnetohydrodynamics code for merger applications, MNRAS, 379 (2007), pp. 915–931.

    Article  Google Scholar 

  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. D. Ryu and T. W. Jones, Numerical magetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow, ApJ, 442 (1995), pp. 228–258.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  62. R. Speith, Untersuchung von Smoothed Particle Hydrodynamics anhand astrophysikalischer Beispiele, PhD thesis, Eberhard-Karls-Universität Tübingen, 1998.

    Google Scholar 

  63. V. Springel, The cosmological simulation code GADGET-2, MNRAS, 364 (2005), pp. 1105–1134.

    Article  Google Scholar 

  64. V. Springel and L. Hernquist, Cosmological smoothed particle hydrodynamics simulations: the entropy equation, MNRAS, 333 (2002), pp. 649–664.

    Article  Google Scholar 

  65. D. Stern, Euler potentials, American Journal of Physics, 38 (1970), p. 494.

    Article  Google Scholar 

  66. D. P. Stern, The Motion of Magnetic Field Lines, Space Science Reviews, 6 (1966), p. 147.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  68. C. Thompson and R. C. Duncan, Neutron star dynamos and the origins of pulsar magnetism, ApJ, 408 (1993), pp. 194–217.

    Article  Google Scholar 

  69. L. M. Widrow, Origin of galactic and extragalactic magnetic fields, Reviews of Modern Physics, 74 (2002), pp. 775–823.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Jacobs University Bremen, Campus Ring 1, D-28759, Bremen, Germany

    Stephan Rosswog

  2. School of Physics, University of Exeter, Stocker Rd, Exeter, EX4 4QL, UK

    Daniel Price

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  1. Stephan Rosswog
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  2. Daniel Price
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Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Wegelerstrasse 6, 53115, Bonn, Germany

    Michael Griebel  & Marc Alexander Schweitzer  & 

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Rosswog, S., Price, D. (2008). 3D Meshfree Magnetohydrodynamics. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79994-8_15

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