Summary
This paper presents the methodology behind and results of adaptive mesh refinement in global magnetohydrodynamic models of the space environment. Techniques used in solving the governing equations of semi-relativistic magnetohydrodynamics (MHD) are presented. These techniques include high-resolution upwind schemes, block-based solution-adaptive grids, explicit, implicit and partial-implicit time-stepping, and domain decomposition for parallelization. Recent work done in coupling the MHD model to upper-atmosphere and inner-magnetosphere models is presented, along with results from modeling a solar coronal mass ejection and its interaction with Earth’s magnetosphere.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. J. Berger. Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations. PhD thesis, Stanford Univ., Stanford, Calif., January 1982.
M. J. Berger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53:484–512, 1984.
M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82:67–84, 1989.
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, June 1989.
M. J. Berger and S. Saltzman. AMR on the CME-2. Appl. Numer. Math., 14:239–253, 1994.
J. P. Boris. A physically motivated solution of the Alfvén problem. Technical Report NRL Memorandum Report 2167, Naval Research Laboratory, Washington, D.C., 1970.
J.U. Brackbill and D.C. Barnes. The effect of nonzero ∇· B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys., 35:426–430, 1980.
W. Dai and P. R. Woodward. A simple finite difference scheme for multidimensional magnetohydrodynamic equations. J. Comput. Phys., 142:331, 1998.
A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg. Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys., 00:00–00, 2001. submitted.
G. M. Erickson, R. W. Spiro, and R. A. Wolf. The physics of the harang discontinuity. J. Geophys. Res., 96:1633–1645, 1991.
S. K. Godunov. Symmetric form of the equations of magnetohydrodynamics (in Russian). In Numerical Methods for Mechanics of Continuum Medium, volume 1, pages 26–34. Siberian Branch of USSR Acad. of Sci., Novosibirsk, 1972.
T. I. Gombosi, G. Tóth, D. L. De Zeeuw, K. C. Hansen, K. Kabin, and K. G. Powell. Semi-relativistic magnetohydrodynamics and physics-based convergence acceleration. J. Comput. Phys., 177:176–205, 2002.
M. Harel, R. A. Wolf, P. H. Reiff, R. W. Spiro, W. J. Burke, F. J. Rich, and M. Smiddy. Quantitative simulation of a magnetospheric substorm 1, Model logic and overview. J. Geophys. Res., 86:2217–2241, 1981.
A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49:357–393, 1983.
R. K. Jaggi and R. A. Wolf. Self-consistent calculation of the motion of a sheet of ions in the magnetosphere. J. Geophys. Res., 78:2842, 1973.
R. Keppens, G. Tóth, M. A. Botchev, and A. van der Ploeg. Implicit and semi-implicit schemes: Algorithms. Int. J. for Num. Meth. in Fluids, 30:335–352, 1999.
T. J. Linde. A Three-Dimensional Adaptive Multifluid MHD Model of the Heliosphere. PhD thesis, Univ. of Mich., Ann Arbor, May 1998.
P. Londrillo and L. Del Aanna. High-order upwind schemes for multidimensional magnetohydrodynamics. Astrophys. J., 530:508–524, 2000.
W. B. Manchester, T. I. Gombosi, I. Roussev, D. L. De Zeeuw, I. V. Sokolov, and K. G. Powell. Thre-dimensional mhd simulation of a flux-rope driven cme. J. Geophys. Res., 109, 2004.
K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report 94-24, Inst. for Comput. Appl. in Sci. and Eng., NASA Langley Space Flight Center, Hampton, Va., 1994.
K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. De Zeeuw. A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys., 154(2):284–309, September 1999.
J. J. Quirk. An Adaptive Grid Algorithm for Computational Shock Hydrodynamics. PhD thesis, Cranfield Inst. of Technol., Cranfield, England, January 1991.
J. J. Quirk and U. R. Hanebutte. A parallel adaptive mesh refinement algorithm. Technical Report 93-63, ICASE, August 1993.
A. D. Richmond, E. C. Ridley, and R. G. Roble. A thermosphere/ionosphere general circulation model with coupled electrodynamics. Geophys. Res. Lett., 19:601–604, 1992.
A. J. Ridley, D. L. De Zeeuw, T. I. Gombosi, and K. G. Powell. Using steady-state MHD results to predict the global state of the magnetosphere-ionosphere system. J. Geophys. Res., 106:30,067–30,076, 2001.
A. J. Ridley, T. I. Gombosi, and D. L. De Zeeuw. Ionospheric control of the magnetosphere: Conductance. J. Geophys. Res., 00:00–00, 2002. in preparation.
A. J. Ridley, T. I. Gombosi, D. L. De Zeeuw, C. R. Clauer, and A. D. Richmond. Ionospheric control of the magnetosphere: Neutral winds. J. Geophys. Res., 00:00–00, 2002. in preparation.
A. J. Ridley, K. C. Hansen, G. Tóth, D. L. De Zueew, T. I. Gombosi, and K. G. Powell. University of Michigan MHD results of the GGCM metrics challenge. J. Geophys. Res., 107:000–000, 2002. in press.
P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43:357–372, 1981.
P. L. Roe and D. S. Balsara. Notes on the eigensystem of magnetohydrodynamics. SIAM J. Appl. Math., 56(1):57–67, February 1996.
G. Tóth. The ∇ · B constraint in shock capturing magnetohydrodynamic codes. J. Comput. Phys., 161:605–652, 2000.
G. Tóth, R. Keppens, and M. A. Botchev. Implicit and semi-implicit schemes in the Versatile Advection Code: Numerical tests. Astron. Astrophys., 332:1159–1170, 1998.
B. van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys., 32:101–136, 1979.
R. A. Wolf. The quasi-static (slow-flow) region of the magnetosphere. In R. L. Carovillano and J. M. Forbes, editors, Solar Terrestrial Physics, pages 303–368. D. Reidel Publishing, Hingham, MA, 1983.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Powell, K.G., De Zeeuw, D.L., Sokolov, I.V., Tóth, G., Gombosi, T.I., Stout, Q. (2005). Parallel, AMR MHD for Global Space Weather Simulations. In: Plewa, T., Linde, T., Gregory Weirs, V. (eds) Adaptive Mesh Refinement - Theory and Applications. Lecture Notes in Computational Science and Engineering, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27039-6_36
Download citation
DOI: https://doi.org/10.1007/3-540-27039-6_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21147-1
Online ISBN: 978-3-540-27039-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)