Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-24T00:10:04.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable upwind schemes for the magnetic induction equation

Published online by Cambridge University Press:  08 April 2009

Franz G. Fuchs ,
Kenneth H. Karlsen ,
Siddharta Mishra  and
Nils H. Risebro
Franz G. Fuchs
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. [email protected]; [email protected]; [email protected]; [email protected]
Kenneth H. Karlsen
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. [email protected]; [email protected]; [email protected]; [email protected]
Siddharta Mishra
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. [email protected]; [email protected]; [email protected]; [email protected]
Nils H. Risebro
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. [email protected]; [email protected]; [email protected]; [email protected]
Get access

Abstract

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.

Keywords

Conservation lawsinduction equationdivergence constraintupwinded source terms
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 825 - 852
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balsara, D.S. and Spicer, D., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comp. Phys. 149 (1999) 270–292. CrossRef
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).
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).
Besse, N. and Kröner, D., Convergence of the locally divergence free discontinuous Galerkin methods for induction equations for the 2D-MHD system. ESAIM: M2AN 39 (2005) 1177–1202. CrossRef
Brackbill, J.U. and Barnes, D.C., The effect of nonzero divB on the numerical solution of the magnetohydrodynamic equations. J. Comp. Phys. 35 (1980) 426–430. CrossRef
Dai, W. and Woodward, P.R., A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331–369. CrossRef
Evans, C. and Hawley, J.F., Simulation of magnetohydrodynamic flow: a constrained transport method. Astrophys. J. 332 (1998) 659. CrossRef
Fuchs, F.G., Mishra, S. and Risebro, N.H., Splitting based finite volume schemes for ideal MHD equations. J. Comp. Phys. 228 (2009) 641–660. CrossRef
Godunov, S.K., The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1 (1972) 26–34.
J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999).
Jovanovič, V. and Rohde, C., Finite volume schemes for Friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Num. Meth. PDE 21 (2005) 104–131.
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002).
G.K. Parks, Physics of Space Plasmas: An Introduction. Addition-Wesley (1991).
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).
Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I. and De Zeeuw, D.L., A solution adaptive upwind scheme for ideal MHD. J. Comp. Phys. 154 (1999) 284–309. CrossRef
J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. Thesis, University of Washington, Seattle, USA (2002).
Ryu, D.S., Miniati, F., Jones, T.W. and Frank, A., A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509 (1998) 244–255. CrossRef
Torrilhon, M., Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166–1191. CrossRef
Torrilhon, M. and Constraint-preserving, M. Fey upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694–1728. CrossRef
Toth, G., The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605–652.
J-P. Vila, P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573–602. CrossRef