An integro-differential formulation for magnetic induction in bounded domains: Boundary element–finite volume method. (English) Zbl 1106.76404
This paper is concerned with the numerical resolution of MHD problems in bounded domains. An integral formulation is used on the boundary where the magnetic field has a global nature, and it is combined with a local discretization inside the domain. It is observed that to couple finite volumes with boundary elements allows a formulation of the global boundary conditions in arbitrary geometries that provides a convenient method for parallel computation. Finally, some examples of magnetic diffusion problems in a sphere as well as in a finite cylinder are presented.
Reviewer: V. D. Sharma (Mumbai)
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
Keywords:
MHD; boundary element method; magnetic diffusion; magnetohydrodynamics; electrohydrodynamics; hydrodynamic problems; hydromagnetic problems; boundary element methodsSoftware:
GeoFEMReferences:
| [1] | Ferriz-Mas, A.; Núñez Jiménez, M., Advances in nonlinear dynamos, (Soward; Ghil, The Fluid Mechanics of Astrophysics and Geophysics Series (2003), Taylor & Francis: Taylor & Francis London) · Zbl 1084.85500 |
| [2] | Bullard, E. C., The stability of a homopolar dynamo, Proc. Cambridge Philos. Soc., 51, 744-760 (1955) |
| [3] | Christensen, U. R.; Aubert, J.; Cardin, P.; Dormy, E.; Gibbons, S.; Glatzmaier, G. A.; Grote, E.; Honkura, H.; Jones, C.; Kono, M.; Matsushima, M.; Sakuraba, A.; Takahashi, F.; Tilinger, A.; Wicht, J.; Zhang, K., A numerical dynamo benchmark, Phys. Earth Planet. Int., 128, 25-34 (2001) |
| [4] | Tilgner, A., Numerical simulation of the onset of dynamo action in an experimental two-scale dynamo, Phys. Fluids, 14, 11 (2002) · Zbl 1185.76368 |
| [5] | Stewartson, K., On almost rigid rotations, J. Fluid Mech., 3, 299-303 (1957) · Zbl 0080.39103 |
| [6] | Proudman, I., The almost-rigid rotation of viscous fluid between concentric spheres, J. Fluid Mech., 1, 505-516 (1956) · Zbl 0072.20104 |
| [7] | Dormy, E.; Jault, D.; Soward, A., A super-rotating shear layer in magnetohydrodynamic spherical Couette flow, J. Fluid Mech., 452, 263-291 (2002) · Zbl 1029.76060 |
| [8] | Taylor, J. B., The magneto-hydrodynamics of a rotating fluid and the Earth’s dynamo problem, Proc. R. Soc. Lond. A, 274, 274-283 (1963) · Zbl 0125.43804 |
| [9] | H. Matsui, H. Okuda, Development of a simulation code for MHD dynamo processes using the GeoFEM platform, Int. J. Comput. Fluid Dynamics (2004), in press; H. Matsui, H. Okuda, Development of a simulation code for MHD dynamo processes using the GeoFEM platform, Int. J. Comput. Fluid Dynamics (2004), in press · Zbl 1221.76230 |
| [10] | Chan, K.; Zhang, K.; Zou, J.; Schubert, G., A non-linear 3D spherical \(α^2\) dynamo using a finite element method, Phys. Earth Planet. Int., 128, 35-50 (2001) |
| [11] | Balsara, D.; Spicer, D., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149, 270-292 (1999) · Zbl 0936.76051 |
| [12] | Balsara, D., Divergence-free adaptive mesh refinement for magneto-hydrodynamics, J. Comput. Phys., 174, 614-648 (2001) · Zbl 1157.76369 |
| [13] | Archontis, V.; Dorch, S. B.F; Nordlund, A., Numerical simulations of kinematic dynamo action, A&A, 397, 393-399 (2003) · Zbl 1069.76057 |
| [14] | Kageyama, A.; Ochi, M.; Sato, T., Flip-flop transitions of the magnetic intensity and polarity reversals in the magnetohydrodynamic dynamo, Phys. Rev. Lett., 62, 26, 5409-5412 (1999) |
| [15] | Rüdiger, G.; Zhang, Y., MHD instability in differentially-rotating cylindric flows, A&A, 378, 302-308 (2001) · Zbl 1038.85003 |
| [16] | Stefani, F.; Gerbeth, G.; Rädler, K.-H, Steady dynamos in finite domains: an integral equation approach, Astron. Nachr., 321, 65-73 (2000) · Zbl 1092.76075 |
| [17] | M. Xu, F. Stefani, G. Gerbeth, The integral equation method for a steady kinematic dynamo problem, J. Comput. Phys. (2004), in press; M. Xu, F. Stefani, G. Gerbeth, The integral equation method for a steady kinematic dynamo problem, J. Comput. Phys. (2004), in press · Zbl 1115.76426 |
| [18] | Jepps, S. A., Numerical models of hydromagnetic dynamos, J. Fluid Mech., 67, part 4, 625-646 (1975) · Zbl 0299.76050 |
| [19] | Ivanova, T. S., A method for solving hydromagnetic dynamo problem, J. Comput. Math. Math. Phys., 16, 4, 956-968 (1976), (in Russian) · Zbl 0363.76080 |
| [20] | Hejda, P., Boundary conditions in the hydromagnetic dynamo problem, Stud. Geophys. Geod., 26, 361-372 (1982) |
| [21] | D’Ippolito, D. A.; Freidberg, J. P.; Goedbloed, J. P.; Rem, J., High-beta tokamaks surrounded by force-free fields, Phys. Fluids, 21, 9, 1600-1616 (1978) · Zbl 0385.76092 |
| [22] | Freidberg, J. P.; Grossmann, W., Magnetohydrodynamic stability of a sharp boundary model of tokamak, Phys. Fluids, 18, 11, 1494-1506 (1975) |
| [23] | Tóth, G., The ∇·\(B=0\) constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 605-652 (2000) · Zbl 0980.76051 |
| [24] | Yee, K., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antenna Propagation, AP-14, 302 (1966) · Zbl 1155.78304 |
| [25] | Samarskii, A. A.; Tishkin, V. F.; Favorskii, A. P.; Shashkov, M. Yu, Operational finite-difference schemes, Differential Equations, 17, 7, 854-862 (1981) · Zbl 0485.65060 |
| [26] | Shashkov, M., Conservative Finite-Difference Methods on General Grids (1996), CRC Press: CRC Press Boca Raton, FL, pp. 47-149 · Zbl 0844.65067 |
| [27] | Dormy, E.; Tarantola, A., Numerical simulation of elastic wave propagation using a finite volume method, J. Geophys. Res., 100, 2123-2133 (1995) |
| [28] | Beer, G., Programming the Boundary Element Method. An Introduction for Engineers (2001), Wiley: Wiley New York |
| [29] | Bonnet, M., Equations intégrales et éléments de frontière (1995), CNRS Editions/Editions Eyrolles |
| [30] | Moffatt, H. K., Magnetic Field Generation in Electrically Conducting Fluids (1978), Cambridge University Press: Cambridge University Press Cambridge, MA · Zbl 0393.76063 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
