Stable upwind schemes for the magnetic induction equation
Fuchs, Franz G. ; Karlsen, Kenneth H. ; Mishra, Siddharta ; Risebro, Nils H.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009), p. 825-852 / Harvested from Numdam
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We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.

Publié le : 2009-01-01
DOI : https://doi.org/10.1051/m2an/2009006
Classification:  65M12,  35L65 
@article{M2AN_2009__43_5_825_0,
     author = {Fuchs, Franz G. and Karlsen, Kenneth H. and Mishra, Siddharta and Risebro, Nils H.},
     title = {Stable upwind schemes for the magnetic induction equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {43},
     year = {2009},
     pages = {825-852},
     doi = {10.1051/m2an/2009006},
     mrnumber = {2559735},
     zbl = {1177.78057},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2009__43_5_825_0}
}

Fuchs, Franz G.; Karlsen, Kenneth H.; Mishra, Siddharta; Risebro, Nils H. Stable upwind schemes for the magnetic induction equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) pp. 825-852. doi : 10.1051/m2an/2009006. http://gdmltest.u-ga.fr/item/M2AN_2009__43_5_825_0/

[1] D.S. Balsara and D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comp. Phys. 149 (1999) 270-292. | MR 1672743 | Zbl 0936.76051

[2] T.J. Barth, Numerical methods for gas dynamics systems, in An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde Eds., Springer (1999). | MR 1731618 | Zbl 0969.76040

[3] S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic, Partial differential equations - First-order systems and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007). | MR 2284507 | Zbl 1113.35001

[4] N. Besse and D. Kröner, Convergence of the locally divergence free discontinuous Galerkin methods for induction equations for the 2D-MHD system. ESAIM: M2AN 39 (2005) 1177-1202. | Numdam | MR 2195909 | Zbl 1084.76046

[5] J.U. Brackbill and D.C. Barnes, The effect of nonzero divB on the numerical solution of the magnetohydrodynamic equations. J. Comp. Phys. 35 (1980) 426-430. | MR 570347 | Zbl 0429.76079

[6] W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331-369. | MR 1624220 | Zbl 0932.76048

[7] C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow: a constrained transport method. Astrophys. J. 332 (1998) 659.

[8] F.G. Fuchs, S. Mishra and N.H. Risebro, Splitting based finite volume schemes for ideal MHD equations. J. Comp. Phys. 228 (2009) 641-660. | MR 2477781 | Zbl pre05506587

[9] S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1 (1972) 26-34.

[10] J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999). | MR 436782 | Zbl 0920.00012

[11] V. Jovanovič and C. Rohde, Finite volume schemes for Friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Num. Meth. PDE 21 (2005) 104-131. | MR 2100303 | Zbl 1068.65113

[12] R.J. Leveque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[13] G.K. Parks, Physics of Space Plasmas: An Introduction. Addition-Wesley (1991).

[14] K.G. Powell, A approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report 94-24, ICASE, Langley, VA, USA (1994).

[15] K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi and D.L. De Zeeuw, A solution adaptive upwind scheme for ideal MHD. J. Comp. Phys. 154 (1999) 284-309. | MR 1712592 | Zbl 0952.76045

[16] J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. Thesis, University of Washington, Seattle, USA (2002).

[17] D.S. Ryu, F. Miniati, T.W. Jones and A. Frank, A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509 (1998) 244-255.

[18] M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166-1191. | MR 2143480 | Zbl 1149.76693

[19] M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694-1728. | MR 2114297 | Zbl 1146.76621

[20] G. Toth, The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605-652. | MR 1764250 | Zbl 0980.76051

[21] J-P. Vila and P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573-602. | MR 1981168 | Zbl 1030.65110