Abstract
It has been believed that three-dimensional, numerical, magnetohydrodynamic (MHD) modelling must play a crucial role in a seamless forecasting system. This system refers to space weather originating on the sun; propagation of disturbances through the solar wind and interplanetary magnetic field (IMF), and thence, transmission into the magnetosphere, ionosphere, and thermosphere. This role comes as no surprise to numerical modelers that participate in the numerical modelling of atmospheric environments as well as the meteorological conditions at Earth. Space scientists have paid great attention to operational numerical space weather prediction models. To this purpose practical progress has been made in the past years. Here first is reviewed the progress of the numerical methods in solar wind modelling. Then, based on our discussion, a new numerical scheme of total variation diminishing (TVD) type for magnetohydrodynamic equations in spherical coordinates is proposed by taking into account convergence, stability and resolution. This new MHD model is established by solving the fluid equations of MHD system with a modified Lax-Friedrichs scheme and the magnetic induction equations with MacCormack II scheme for the purpose of developing a combined scheme of quick convergence as well as of TVD property. To verify the validation of the scheme, the propagation of one-dimensional MHD fast and slow shock problem is discussed with the numerical results conforming to the existing results obtained by the piece-wise parabolic method (PPM). Finally, some conclusions are made.
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Brackbill, J. U. and Barnes, D. C.: 1980, J.Comput.Phys. 35, 426-430.
Brecht, S. H. et al.: 1982, J.Geophys.Res. 87, 6098-6108.
Burlaga, L. F., Plunkett, S. P., St. Cyr, O. C.: 2002, J.Geophys.Res. 107(A10), 1266, doi:10.1029/2001JA000255.
Chao, J. K. and Chen, H. H.: 2001, ‘Prediction of Southward IMF B z’, in P. Song, H. J. Singer and G. L. Siscoe (eds), Space Weather, Geophysical Monograph 125, AGU Washington, pp. 143-158.
Dai, W. and Woodward, P. R.: 1994, J.Comput.Phys. 115, 485-514.
Dryer, M.: 1998, AIAA Journal 36, 365-390.
De Zeeuw, D. L. et al.: 2000, IEEE Trans.Plasma Sci. 28, 1956-1965.
Evans, C. R. and Hawley, J. F.: 1988, Astrophys.J. 332, 659-677.
Fedder, J. A., Linker, S. P. and Lyon, J. G.: 1998, J.Geophys.Res. 103A, 14799-14810.
Godunov S. K.: 1972, ‘Symmetric Form of magnetohydrodynamics (in Russian)’, in Numerical Methods for Mechanics of Continium Medium, Vol. 1, Siberian Branch of USSR Acad. of Sci., pp. 26-34.
Gombosi, T. I. et al.: 2000a, Adv.Space Res. 26(1), 139-149.
Gombosi, T. I. et al.: 2000b, J.Atmos.Solar-Terr.Phys. 62(16), 1515-1525.
Gombosi, T. I. et al.: 2001, ‘From Sun to Earth: Multiscale MHD simulation of space weather’, in P. Song, H. J. Singer, and G. L. Siscoe (eds), Space Weather, Geophysical Monograph 125, AGU Washington, pp. 169-176.
Groth, C. P. T. et al.: 1999, Space Sci.Rev. 87, 193-198.
Groth, C. P. T. et al.: 2000a, Adv.Space Res. 26(5), 793-800.
Groth, C. P. T. et al.: 2000b, J.Geophys.Res. 105, 25053-25078.
Guo, W. P. and Wu, S. T.: 1998, Astrophys.J. 494, 419-429.
Han, S. M., Wu, S. T. and Dryer, M.: 1988, Comp.Fluids 16, 81-103.
Harned, D. S. and Kerner, W.: 1985, J.Comput.Phys. 60, 62-75.
Hu, Y. Q. and Wu, S. T.: 1984, J.Comput.Phys. 55, 33-64.
Janhunen, P.: 1996, ‘GUMICS-3: A Global Ionosphere-Magnetosphere Coupling Simulation with High Ionospheric Resolution’, in Proc. ESA 1996 Symposium on Environment Modelling for Space-based Applications, ESA SP-392, pp. 233-239.
Klimchuk, J. A.: 2001, ‘Theory of Coronal mass ejections’, in P. Song, H. J. Singer and G. L. Siscoe (eds), Space Weather, Geophysical Monograph 125, AGU Washington, pp. 143-158.
Kulikovskiy, A. and Lyubimov, G.: 1965, Magnetohydrodynamics. Addison-Wesley, Reading, MA.
Lax, P. D. and Wendroff, B.: 1960, Comm.Pure Appl.Math. 13, 217-237.
Leboeuf, J. N., Tajima, T., Kennel, C. F. and Dawson, J. M.: 1981, Geophys.Res.Lett. 8, 257-260.
Linker, J. A. et al.: 1999, J.Geophys.Res. 104, 9809-9830.
Lionello, R. Z., Mikic, Z. and Linker, J. A.: 1998, J.Comput.Phys. 140, 172-201.
Lyon, J. G., Fedder, J. A. and Huba, J. G.: 1986, J.Geophys.Res. 91, 8057-8064.
Mikic, Z. et al.: 1999, Phys.Plasmas 6, 2217-2224.
Nakagawa, Y., Hu, Y. Q. and Wu, S. T.: 1987, Astron.Astrophys. 179, 354-370.
Odstrcil, D. and Pizzo, V. J.: 1999a, J.Geophys.Res. 104, 483-492.
Odstrcil, D. and Pizzon, V. J.: 1999b, J.Geophys.Res. 104, 493-503.
Odstrcil, D. and Pizzo, V. J.: 1999c, J.Geophys.Res. 104, 28225-28239.
Ogino, T.: 1986, J.Geophys.Res. 91, 6791-6806.
Panitchob, S., Wu, S. T. and Suess, S. T.: 1987, AIAA Paper, 87-1218-CP.
Powell, K. G. et al.: 1995, AIAA Paper, 95-1704-CP.
Powell, K. G. et al.: 1999, J.Comput.Phys. 154, 284-309.
Raeder, J., Walker, J. and Ashour-Abdalla, M.: 1998, J.Geophys.Res. 103, 14787-14797.
Ridley, A. J. et al.: 2002, J.Geophys.Res. 107(A10), 1290, doi: 10.1029/2001JA000253.
Riley, P. et al.: 1997, Adv.Space Res. 20, 15-22.
Riley, P., Gosling, J. T., McComas, D. J. et al.: 1999, J.Geophys.Res. 104, 9871-9879.
Riley, P. et al.: 2001, ‘MHD Modeling of the Solar Corona and Inner Heliosphere: Comparison with Observations’, in P. Song, H. J. Singer, and G. L. Siscoe (eds), Space Weather, Geophysical Monograph 125, AGU Washington, pp. 159-168.
Riley, P., Linker, J. A. and Mikic, Z.: 2002, J.Geophys.Res. 107(A7), doi: 10.1029/2001JA000299.
Shi, Y., Wei, F. S. and Feng, X. S.: 2001, Science in China (A) 30, 61-64.
Song, P. et al.: 1999, J.Geophys.Res. 104, 28,361-28,378.
Song, P. et al.: 2000, Planetary Space Sci. 48, 29-39.
St Cyr, O. C. et al.: 2000, J.Geophys.Res. 105, 12493-12506.
Steinolfson, R. S.: 1991, Astrophys.J. 382, 677-687.
Steinolfson, R. S.: 1994, Space Sci.Rev. 70, 289-294.
Tanaka, T.: 1994, J.Comput.Phys. 111, 381-389.
Tanaka, T.: 1995, J.Geophys.Res. 100, 12057-12074.
Toth, G. and Odstrcil, D.: 1996, J.Comput.Phys. 128, 82-100.
Toth, G.: 2000, J.Comput.Phys. 161, 605-652.
Vandas, M., Odstrcil, D. and Watari, S.: 2002, J.Geophys.Res. 107(A9), 1236, doi: 10.1029/2001JA005068.
Wang, A. H., Wu, S. T., Suess, S. T. and Poletto, G.: 1998, J.Geophys.Res. 103, 1913-1922.
Washimi, H.: 1990, Geophys.Res.Lett. 17, 33-40.
Watanabe, K. and Sato, T.: 1990, J.Geophys.Res. 95, 13437-13454.
Webb, D. R. et al.: 2001, ‘The Solar Sources of Geoeffective Structures’, in P. Song, H. J. Singer and G. L. Siscoe (eds), Space Weather, Geophysical Monograph 125, AGU Washington, pp. 123-143.
White, W. W. et al.: 1998, Geophys.Res.Lett. 25, 1605-1608.
Winglee, R. M.: 1994, J.Geophys.Res. 99, 13437-13454.
Wu, C. C., Walker, R. J. and Dawson, J. M.: 1981, Geophys.Res.Lett. 8, 523-526.
Wu, C. C., Dryer, M. et al.: 1996a, J.Atmos.Solar-Terr.Phys. 58, 1805-1812.
Wu, C. C. and Dryer, M.: 1996b, Geophys.Res.Lett. 23, 1709-1712.
Wu, C. C. and Dryer, M.: 1997, Solar Phys. 173, 391-408.
Wu, S. T., Andrews, M. D. and Plunkett, S. P.: 2001, Space Sci.Rev. 95, 191-213.
Wu, S. T., Hu, Y. Q., Nakagawa, Y. and Tandberg-Hanssen, E.: 1986, Astrophys.J. 306, 751-761.
Wu, S. T. and Wang, A. H.: 1991, Adv.Space Res. 11, 187-204.
Wu, S. T., Guo, W. P. and Wang, J. F.: 1995, Solar Phys. 157, 325-348.
Wu, S. T., Guo, W. P. et al.: 1997, Solar Phys. 175, 719-735.
Wu, S. T., Guo, W. P. and Dryer, M.: 1997, Solar Phys. 170, 265-282.
Wu, S. T., Guo, W. P. et al.: 1999, J.Geophys.Res. 104, 14789-14802.
Wu, S. T. et al.: 2000a, Astrophys.J. 545, 1101-1115.
Wu, S. T. et al.: 2000b, J.Atmos.Solar-Terr.Phys. 62(16), 1489-1498.
Yee, H. C.: 1989, A Class of High Resolution Explicit and Implicit Shock-Capturing Methods. NASA TM-101088.
Zhang, B. C. and Wang, J. F.: 2000, J.Geophys.Res. 105, 12593-12603.
Zhang, J. H. and Wei, F. S.: 1993, Science in China(A) 23(4), 427-436.
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Feng, X., Wu, S., Wei, F. et al. A Class of TVD Type Combined Numerical Scheme for MHD Equations With a Survey About Numerical Methods in Solar Wind Simulations. Space Science Reviews 107, 43–53 (2003). https://doi.org/10.1023/A:1025547016708
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DOI: https://doi.org/10.1023/A:1025547016708