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Adaptive Mesh Refinement for Global Magnetohydrodynamic Simulation

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Space Plasma Simulation

Part of the book series: Lecture Notes in Physics ((LNP,volume 615))

Abstract

The first part of this paper reviews some physics issues representing major computational challenges for global MHD models of the space environment. These issues include: (i) mathematical formulation and discretization of the governing equations that ensure the proper jump conditions and propagation speeds, (ii) regions of relativistic Alfvén speed, (iii) regions dominated by strong intrinsic planetary magnetic field with strong gradients, and (iv) the religiously debated issue of controling the divergence of the magnetic field. The second part of the paper concentrates to modern solution methods that have been developed by the aerodynamics, applied mathematics and DoE communities. Such methods have recently begun to be implemented in space-physics codes, which solve the governing equations for a compressible magnetized plasma. These techniques include high-resolution upwind schemes, block-based solution-adaptive grids and domain decomposition for parallelization. While some of these techniques carry over relatively straightforwardly to space physics, space physics simulations pose some new challenges. We give a brief review of the state-of-the-art in modern space-physics codes. Finally, we describe the space physics MHD code developed at the University of Michigan and its recent coupling to a thermosphere-ionosphere and inner magnetosphere model.

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References

  1. D. S. Balsara and D. S. Spicer: J. Comput. Phys., 149, 270 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. D. S. Balsara and D. S. Spicer: J. Comput. Phys., 148, 133 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. D. Balsara: J. Comput. Phys., 174, 614 (2001)

    Article  MATH  ADS  Google Scholar 

  4. R. Bauske, A. F. Nagy, T. I. Gombosi, D. L. De Zeeuw, K. G. Powell and J. G. Luhmann: J. Geophys. Res., 103, 23,625 (1998)

    Article  ADS  Google Scholar 

  5. M. J. Berger: ‘Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations’. Interanl report at Stanford Univ., Stanford, Calif., 1982

    Google Scholar 

  6. M. J. Berger: J. Comput. Phys., 53, 484 (1984)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. M. J. Berger and A. Jameson: AIAA Journal, 32, 561 (1985)

    Article  ADS  Google Scholar 

  8. M. J. Berger and P. Colella: J. Comput. Phys., 82, 67 (1989)

    Article  ADS  Google Scholar 

  9. M. J. Berger and R. J. LeVeque: ‘An adaptive Cartesian mesh algorithm for the Euler equations in arbitrary geometries’. In Proc. 9th AIAA Computational Fluid Dynamics Conference (AIAA Paper No. 89-1930, Buffalo, NY 1989)

    Google Scholar 

  10. M. J. Berger and S. Saltzman: “Appl. Numer. Math.” 14, 239 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Boris, J. P.: ‘A physically motivated solution of the Alfvén problem’. Internal report at Naval Research Laboratory, NRL Memorandum Report 2167,Washington, D.C., 1970

    Google Scholar 

  12. J.U. Brackbill and D.C. Barnes: J. Comput. Phys., 35, 426 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. W. Dai and P. R. Woodward: J. Comput. Phys., 142, 331 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Dedner, A., F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer and M. Wesenberg: J. Comput. Phys., 175, 645 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. H. De Sterck: ‘Multi-Dimensional Upwind Constrained Transport on Unstructured Grids for’ shallow Water’ Magnetohydrodynamics’. In Proceedings of AIAA 15th Computational Dynamics Conference, (AIAA-2001-2623, 2001)

    Google Scholar 

  16. D. L. De Zeeuw and K. G. Powell: J. Comput. Phys., 104, 55 (1993)

    Article  Google Scholar 

  17. D. L. De Zeeuw, S. Sazykin, A. J. Ridley, G. Tóth, T. I. Gombosi, K. G. Powell and R. A. Wolf: J. Geophys. Res., (2002) “in preparation”

    Google Scholar 

  18. Erickson, G. M., R. W. Spiro and R. A. Wolf: J. Geophys. Res., 96, 1633 (1991)

    Article  ADS  Google Scholar 

  19. C. R. Evans and J. F. Hawley: Astrophys. Journ., 332, 659 (1988)

    Article  ADS  Google Scholar 

  20. Fedder, J. A., S. P. Slinker, J. G. Lyon and R. D. Elphinstone: J. Geophys. Res., 100, 19,083 (1995)

    ADS  Google Scholar 

  21. S. K. Godunov: ‘Symmetric form of the equations of magnetohydrodynamics’ (in Russian). In: Numerical Methods for Mechanics of Continuum Medium, “Siberian Branch of USSR Acad. of Sci.”, 1, 26, Novosibirsk, 1972

    Google Scholar 

  22. T. I. Gombosi, D. L. De Zeeuw, R. M. Häberli and K. G. Powell: J. Geophys. Res., 101, 15233 (1996)

    Article  ADS  Google Scholar 

  23. T. I. Gombosi, D. L. De Zeeuw, C. P. T. Groth, K. G. Powell and P. Song: The length of the magnetotail for northward IMF: Results of 3D MHD simulations, In Physics of Space Plasmas ed. by T. Chang and J. R. Jasperse, (MIT Press, Cambridge, Mass., 1998) pp. 121–128

    Google Scholar 

  24. T. I. Gombosi, D. L. De Zeeuw, C. P. T. Groth, K. G. Powell and Q. F. Stout: J. Atmos. Solar-Terr. Phys., 62, 1515 (2000)

    Article  ADS  Google Scholar 

  25. T. I. Gombosi, G. Tóth, D. L. De Zeeuw, K. C. Hansen, K. Kabin and K. G. Powell: J. Comput. Phys., 177, 176 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. C. P. T. Groth, D. L. De Zeeuw, T. I. Gombosi and K. G. Powell: J. Geophys. Res., 105, 25,053 (2000)

    Article  ADS  Google Scholar 

  27. R. M. Häberli, T. I. Gombosi, D. L. DeZeuuw, M. R. Combi and K. G. Powell: Science, 276, 939 (1997)

    Article  ADS  Google Scholar 

  28. K. C. Hansen, T. I. Gombosi, D. L. DeZeeuw, C. P. T. Groth and K. G. Powell: Adv. Space Res., 26, 1681 (2000)

    Article  ADS  Google Scholar 

  29. Harel, M., R. A. Wolf, P. H. Reiff, R. W. Spiro, W. J. Burke, F. J. Rich and M. Smiddy: J. Geophys. Res., 86, 2217 (1981)

    Article  ADS  Google Scholar 

  30. A. Harten: J. Comput. Phys., 49, 357 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. R. K. Jaggi and R. A. Wolf: J. Geophys. Res., 78, 2842 (1973)

    Article  ADS  Google Scholar 

  32. P. Janhunen: J. Comput. Phys., 160, 649 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. K. Kabin, T. I. Gombosi, D. L. DeZeeuw, K. G. Powell and P. L. Israelevich: J. Geophys. Res., 104, 2451 (1999)

    Article  ADS  Google Scholar 

  34. K. Kabin, T. I. Gombosi, D. L. De Zeeuw and K. G. Powell: “Icarus”, 143, 397 (2000)

    Article  ADS  Google Scholar 

  35. K. Kabin, M. R. Combi, T. I. Gombosi, D. L. De Zeeuw, K. C. Hansen and K. G. Powell: Planet. Space Sci., 49, 337 (2001)

    Article  ADS  Google Scholar 

  36. R. Keppens, G. Tóth, M. A. Botchev and A. van der Ploeg: “Int. J. for Num. Meth. in Fluids”, 30, 335 (1999)

    Article  MATH  ADS  Google Scholar 

  37. P. D. Lax and B. Wendroff: Systems of Conservation Laws, “Communications on Pure and Applied Mathematics”, 13, 217–237 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  38. T. J. Linde, T. I. Gombosi, P. L Roe, K. G. Powell and D. L. De Zeeuw: J. Geophys. Res., 103, 1889 (1998)

    Article  ADS  Google Scholar 

  39. T. J. Linde: A Three-Dimensional Adaptive Multifluid MHD Model of the Heliosphere”, school = “Univ. of Mich.”, Ann Arbor, month = may, 1998

    Google Scholar 

  40. Y. Liu, A. F. Nagy, C. P. T. Groth, D. L. De Zeeuw, T. I. Gombosi and K. G. Powell: Geophys. Res. Lett., 26, 2689 (1999)

    Article  ADS  Google Scholar 

  41. B. Marder: J. Comput. Phys., 68, 48 (1987)

    Article  MATH  ADS  Google Scholar 

  42. T. Ogino and R. J. Walker: Geophys. Res. Lett., 11, 1018 (1984)

    Article  ADS  Google Scholar 

  43. H. Paillère, K. G. Powell and D. L. De Zeeuw: ‘A wave-model based refinement criterion for adaptive-grid computation of compressible flows’. In: 30th AIAA Aerospace Sciences Meeting (AIAA-92-0322, Reno, Nevada, 1992)

    Google Scholar 

  44. K. G. Powell, P. L. Roe and J. Quirk: ‘Adaptive-mesh algorithms for computational fluid dynamics’. In: Algorithmic Trends in Computational Fluid Dynmaics ed. by M. Y. Hussaini, A. Kumar and M. D. Salas (SPRINGER-VERLAG, New York, 1993) pp. 303–337

    Google Scholar 

  45. K. G. Powell: ‘An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension)’. Internal report at Inst. for Comput. Appl. in Sci. and Eng., NASA Langley Space Flight Center, Hampton, Va., 94–24, 1994

    Google Scholar 

  46. K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi and D. L. De Zeeuw: J. Comput. Phys., 154, 284 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  47. J. J. Quirk: “An Adaptive Grid Algorithm for Computational Shock Hydrodynamics’. Internal report at Cranfield Inst. of Technol., Cranfield, England”, 1991

    Google Scholar 

  48. J. J. Quirk and U. R. Hanebutte: ‘A parallel adaptive mesh refinement algorithm’. Interanl; report at ICASE, 93–63, 1993

    Google Scholar 

  49. J. Raeder, R. J. Walker and M. Ashour-Abdalla: Geophys. Res. Lett., 22, 349 1995)

    Article  ADS  Google Scholar 

  50. Richmond, A. D., C. Ridley and R. G. Roble: Geophys. Res. Lett., 19, 601 (1992)

    Article  ADS  Google Scholar 

  51. Richmond, A. D.: J. Geomagn. Geoelectr., 47, 191 (1995)

    Google Scholar 

  52. A. J. Ridley, D. L. De Zeeuw, T. I. Gombosi and K. G. Powell: J. Geophys. Res., 106, 30,067 (2001)

    Article  ADS  Google Scholar 

  53. A. J. Ridley, K. C. Hansen, G. Tóth, D. L. De Zueew, T. I. Gombosi and K. G. Powell: J. Geophys. Res., 107 (2002) “in press”

    Google Scholar 

  54. A. J. Ridley, T. I. Gombosi, D. L. De Zeeuw, C. R. Clauer and A. D. Richmond: J. Geophys. Res., 107 (2002) “in preparation”

    Google Scholar 

  55. A. J. Ridley, T. I. Gombosi and D. L. De Zeeuw: J. Geophys. Res. 107 (2002) “in reparation”

    Google Scholar 

  56. R.M. Robinson, R.R. Vondrak, K. Miller, T. Dabbs and D.A. Hardy: J. Geophys. Res., 92, 2565 (1987)

    Article  ADS  Google Scholar 

  57. R. G. Roble and E. C. Ridley: Ann. Geophys., 5A, 369 (1987)

    ADS  Google Scholar 

  58. P. L. Roe: J. Comput. Phys., 43, 357 (1981)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  59. P. L. Roe and D. S. Balsara: SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS J. APPL. MATH., 56, 57 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  60. S. Sazykin and R. A. Wolf and R. W. Spiro and T. I. Gombosi and D. L. De Zeeuw and M. F. Thomsen: Geophys. Res. Lett., (2002) “in preparation”

    Google Scholar 

  61. P. Song, T.I. Gombosi, D.L. De Zeeuw, K.G. Powell and C. P. T. Groth: Planet. Space Sci., 48, 29 (2000)

    Article  ADS  Google Scholar 

  62. P. Song, D. L. De Zeeuw, T. I. Gombosi, C. P. T. Groth and K. G. Powell: J. Geophys. Res., 104, 28,361 (1999)

    ADS  Google Scholar 

  63. J. M. Stone and M. L. Norman: Astrophys. J. Suppl., 80, 791 (1992)

    Article  ADS  Google Scholar 

  64. T. Tanaka: J. Comput. Phys., 111, 381 (1994)

    Article  MATH  ADS  Google Scholar 

  65. G. Tóth, R. Keppens and M. A. Botchev: Astron. Astrophys., 332, 1159 (1998)

    ADS  Google Scholar 

  66. G. Tóth: J. Comput. Phys., 161, 605 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  67. G. Tóth and P. L. Roe: J. Comput. Phys., (2001) “submitted”

    Google Scholar 

  68. G. Tóth, D. L. De Zeeuw, T. I. Gombosi, K. C. Hansen, K. G. Powell, A. J. Ridley, P. L. Roe and I. V. Sokolov: J. Geophys. Res., 107(2002) “in preparation”

    Google Scholar 

  69. B. van Leer: J. Comput. Phys., 32, 101, (1979)

    Article  ADS  Google Scholar 

  70. V. M. Vasyliunas: ‘Mathematical models of magnetospheric convection and its coupling to the ionosphere’. In: Particles and fields in the magnetosphere ed. by B. M. McCormack (D. Reidel Publishing, Dordrecht, Holland 1970) pp. 60–71

    Google Scholar 

  71. R. A. Wolf: ‘The quasi-static (slow-flow) region of the magnetosphere’. In: Solar Terrestrial Physics, ed. by R. L. Carovillano and J. M. Forbes, (D. Reidel Publishing, Hingham, MA 1983) pp. 303–368

    Google Scholar 

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Author information

Authors and Affiliations

  1. Center for Space Environment Modeling, The University of Michigan, Michigan, Ann Arbor, USA

    Tamas I. Gombosi, Darren L. De Zeeuw, Kenneth G. Powell, Aaron J. Ridley, Igor V. Sokolov, Quentin F. Stout & Gábor Tóth

Authors
  1. Tamas I. Gombosi
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  2. Darren L. De Zeeuw
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  3. Kenneth G. Powell
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  4. Aaron J. Ridley
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  5. Igor V. Sokolov
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  6. Quentin F. Stout
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  7. Gábor Tóth
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Editor information

Editors and Affiliations

  1. Max-Plack-Institut für Aeronomie, Max-Planck Str.2, 37191, Katlenburg-Lindau, Germany

    Jörg Büchner

  2. Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, 85741, Garching, Germany

    Manfred Scholer  & Christian T. Dum  & 

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Gombosi, T.I. et al. (2003). Adaptive Mesh Refinement for Global Magnetohydrodynamic Simulation. In: Büchner, J., Scholer, M., Dum, C.T. (eds) Space Plasma Simulation. Lecture Notes in Physics, vol 615. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36530-3_12

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  • DOI: https://doi.org/10.1007/3-540-36530-3_12

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