Stable upwind schemes for the magnetic induction equation

Franz G. Fuchs; Kenneth H. Karlsen; Siddharta Mishra; Nils H. Risebro

doi:10.1051/m2an/2009006

All issues

Volume 43 / No 5 (September-October 2009)

ESAIM: M2AN, 43 5 (2009) 825-852

References

Free Access

Issue		ESAIM: M2AN Volume 43, Number 5, September-October 2009


Page(s)		825 - 852
DOI		https://doi.org/10.1051/m2an/2009006
Published online		08 April 2009

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. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
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). [Google Scholar]
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). [Google Scholar]
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. [CrossRef] [EDP Sciences] [Google Scholar]
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. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331–369. [NASA ADS] [CrossRef] [Google Scholar]
C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow: a constrained transport method. Astrophys. J. 332 (1998) 659. [NASA ADS] [CrossRef] [Google Scholar]
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. [CrossRef] [Google Scholar]
S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1 (1972) 26–34. [Google Scholar]
J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999). [Google Scholar]
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. [Google Scholar]
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002). [Google Scholar]
G.K. Parks, Physics of Space Plasmas: An Introduction. Addition-Wesley (1991). [Google Scholar]
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). [Google Scholar]
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. [NASA ADS] [CrossRef] [Google Scholar]
J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. Thesis, University of Washington, Seattle, USA (2002). [Google Scholar]
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. [NASA ADS] [CrossRef] [Google Scholar]
M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166–1191. [CrossRef] [Google Scholar]
M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694–1728. [CrossRef] [MathSciNet] [Google Scholar]
G. Toth, The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605–652. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
J-P. Vila and P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573–602. [CrossRef] [MathSciNet] [Google Scholar]

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[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. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[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). [Google Scholar]

[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). [Google Scholar]

[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. [CrossRef] [EDP Sciences] [Google Scholar]

[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. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[6] W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331–369. [NASA ADS] [CrossRef] [Google Scholar]

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

[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. [CrossRef] [Google Scholar]

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

[10] J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999). [Google Scholar]

[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. [Google Scholar]

[12] R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002). [Google Scholar]

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

[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). [Google Scholar]

[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. [NASA ADS] [CrossRef] [Google Scholar]

[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). [Google Scholar]

[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. [NASA ADS] [CrossRef] [Google Scholar]

[18] M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166–1191. [CrossRef] [Google Scholar]

[19] M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694–1728. [CrossRef] [MathSciNet] [Google Scholar]

[20] G. Toth, The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605–652. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[21] J-P. Vila and P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573–602. [CrossRef] [MathSciNet] [Google Scholar]